To complement Justin's concise answer as it is lacking in details, no the Valemen did not immediately give up fighting. They had in fact inflicted one of the first major setbacks on the Targaryen invading forces in the Naval battle off Gulltown, defeating the first Targaryen attempt at conquering Vale.
Timeline of Conquest of Vale
First Diplomatic bout
When the Targaryen invasion began, Vale was ruled by Dowager Queen Sharra Arryn who was regent of her underage son, King Ronnel Arryn. When Aegon landed on Westerosi shores, he soon received an offer from the Queen Regent of Vale:
Another offer of alliance came from the boy king of the Eyrie, Ronnel
Arryn, whose mother asked for all the lands east of the Green Fork of
the Trident for the Vale’s support against Black Harren.
The World of Ice and Fire - The Conquest
So at that time, Valemen probably did not realize the full danger that they were facing. They thought the war would probably end in Riverlands and they could make a profit off it through an alliance with the Targaryens.
First Invasion attempt
Aegon of course, naturally rejected the proposal as he meant to take all of Westeros as subjects, he would suffer no other Kings, let alone have them demand lands for an alliance. He sent his answer to Vale in shape of his Warships and Vhagar.
The Targaryen fleet, under Daemon Velaryon, left Blackwater Bay and
turned north, for Gulltown and the Vale. With them went Queen Visenya
and Vhagar.
The World of Ice and Fire - The
Conquest
The result of this bold move was not entirely what Aegon had anticipated:
A hastily assembled Arryn fleet, augmented by a dozen Braavosi
warships, met and defeated the Targaryen fleet in the waters off
Gulltown. Amongst the dead was Aegon’s admiral, Daemon
Velaryon.
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Conquest
The only things that saved them from an absolute disaster were Queen Visenya and her dragon Vhagar.
Such defeats proved no more than setbacks, however, and in the end,
Aegon’s enemies had no answer for his dragons. The men of the Vale
sank a third of the Targaryen ships and captured near as many, but
when Queen Visenya descended upon them from the sky, their own ships
burned.
The World of Ice and Fire - The Conquest
Pictured below, Queen Visenya burning Valeman fleet:
Second Diplomatic Bout
After the ill-fated Naval invasion of Vale, Aegon found himself preoccupied with more immediate foes in Shape of King Argilac of Stormlands and King Harren of Rivers and Isles. Aegon prevailed against Harren, while Orys Baratheon and Queen Rhaenys subdued the Stormlands.
With two Kingdoms fallen to the Conqueror, Remaining Kings finally realized their peril. Queen Regent Sharra Arryn, sensing that her own turn was coming soon, took following steps:
- She assembled her armies and sent one to the Bloody Gates.
- She took refuge in the Stronghold of Eyrie.
- She ordered strengthening of the Defenses.
When she was as prepared as she could hope to be for battle, she decided to give another try to diplomacy, this time, a lot more sweeter than it was previously, but still demanding something huge.
In her youth Queen Sharra had been lauded as “the Flower of the
Mountain,” the fairest maid in all the Seven Kingdoms. Perhaps hoping
to sway Aegon with her beauty, she sent him a portrait of herself and
offered herself to him in marriage, provided he named her son Ronnel
as his heir. Though the portrait did finally reach him, it is not
known whether Aegon Targaryen ever replied to her proposal; he had two
queens already, and Sharra Arryn was by then a faded flower, ten years
his elder.
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Conquest
As she was once known as most fairest maiden in Westeros, she had hoped to sway Aegon with her beauty. Naturally Aegon dismissed this offer as well since what is the whole point of conquering if you don't leave your titles for your Kids to inherit? Plus what was to be gained if he was to agree? A wife? Aegon already had two, who were younger than the one offered and presumably, prettier as well.
Second Invasion
Aegon again waited until he had dealt with Allied Armies of High Garden and Casterly Rock and subdued the Northmen. Now that his hands were free again, he decided to complete the invasion. He charged Rhaenys with conquest of Dorne, took the task of taking Old Town himself while charging Visenya with conquest of Vale.
Now once again Aegon Targaryen and his queens parted company. Aegon
turned south once more, marching toward Oldtown, whilst his two
sisters mounted their dragons—Visenya for a second attempt at the Vale
of Arryn, and Rhaenys for Sunspear and the deserts of Dorne.
The World of Ice and Fire - The Conquest
In the meanwhile, Valemen were ready to receive the blow.
Sharra Arryn had strengthened the defenses of Gulltown, moved a strong
host to the Bloody Gate, and tripled the size of the garrisons in
Stone, Snow, and Sky, the waycastles that guarded the approach to the
Eyrie.
The World of Ice and Fire - The Conquest
Queen Visenya, showed the futility of these preparations when she landed directly in the Courtyard of Eyrie, finding herself face to face with the young King Ronnel Arryn.
All these defenses proved useless against Visenya Targaryen, who rode
Vhagar’s leathery wings above them all and landed in the Eyrie’s inner
courtyard.
When the regent of the Vale rushed out to confront her, with a dozen
guards at her back, she found Visenya with Ronnel Arryn seated on her
knee, staring at the dragon, wonder-struck.
“Mother, can I go flying with the lady?” the boy king asked. No
threats were spoken, no angry words exchanged. The two queens smiled
at one another and exchanged courtesies instead.
Then Lady Sharra sent for the three crowns (her own regent’s coronet,
her son’s small crown, and the Falcon Crown of Mountain and Vale that
the Arryn kings had worn for a thousand years), and surrendered them
to Queen Visenya, along with the swords of her garrison.
And it was said afterward that the little king flew thrice about the
summit of the Giant’s Lance and landed to find himself a little lord.
Thus did Visenya Targaryen bring the Vale of Arryn into her brother’s
realm
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Pictured, Shadow of Vhagar on Walls of Eyrie while King Ronnel and his mother watch awestruck.
You have to applaud the tact Queen Visenya showed here.
She was courteous and Kind to the young King, promising him a ride on her dragon even. That showed the Valemen, that if they surrendered, Targaryens aren't some savages who would slaughter them or oppress them just for the heck of it. They were humans....with dragons. There was also an unspoken threat, "If I can land right within your courtyard and find your King, I can just as easily burn him to a crisp from the air. None of your defenses will help you. Remember Harrenhal?".
Queen Sharra was quick to see both the carrot and the stick and made the wise decision to surrender to the Targaryens. Thus Queen Visenya won a bloodless victory and Former-King-Later-Lord-Paramount Ronnel got a ride on Vhagar, probably the only non-Valyrian to boast of that other than Alys Rivers.
Since that day till the reign of Mad King Aerys II, Arryns remained loyal vassals to House Targaryen.
The Arryns played their part in the wars of the Targaryen kings, and
in the Blackfyre rebellions,standing stoutly with the Iron Throne
against the Blackfyre Pretenders. During the First Blackfyre
Rebellion, Lord Donnel Arryn boldly led the vanguard of the royalist
host, though his lines were shattered by Daemon Blackfyre, and his
lordship in peril for his life until Ser Gwayne Corbray of the
Kingsguard appeared with reinforcements.
The World of Ice
and Fire - House Arryn
They held a very special place in the Targaryen vassals, honored with providing a bride for King Viserys I and having a Targaryen princess as bride for Lord Rodrik Arryn, something which only House Martell can boast of other than Arryns.
House Arryn can even boast the rare distinction of twice being
deemed worthy of marriage with the blood of the dragon.
Rodrik Arryn, Lord of the Eyrie, was honored by King Jaehaerys I Targaryen and his wife, the Good Queen Alysanne, with the hand of
their daughter, Princess Daella, and a child of that union, the Lady
Aemma Arryn, in turn became the first wife of King Viserys I Targaryen and mother to his firstborn child, Princess Rhaenyra,
who contended with her half brother Aegon II for the Iron Throne.
In that struggle, Jeyne Arryn, Lady of the Eyrie and Maiden of the
Vale, proved a staunch friend to Rhaenyra Targaryen and her sons,
ultimately serving as one of the regents for King Aegon III.
From that day, every Targaryen to sit the Iron Throne had a bit of
Arryn blood.
The World of Ice
and Fire - House Arryn
From that day here means the day Rhaenyra Targaryen was born. Rhaenyra was daughter of Queen Aemma Arryn(Herself a half-Targaryen being grand-daughter of King Jaehaerys I and first cousin to her husband) and King Viserys I Targaryen. All of the Targaryen monarchs after Aegon II were descended from Rhaenyra's sons King Aegon III and King Viserys II so therefore, all of them had Arryn blood in their veins.
How well would natural defenses hold against Dragons?
Not very well. Natural defenses can thwart only land-based attacks. They cannot do anything to someone attacking from the air. As you noted yourself, Harrenhal is the prime example, even if you discount the aerial bloodless quasi-invasion of Vale by Queen Visenya.
Quoted, the conversation between King Aegon and King Harren:
“Yield now,” Aegon began, “and you may remain as Lord of the Iron
Islands. Yield now, and your sons will live to rule after you. I have
eight thousand men outside your walls.”
“What is outside my walls is of no concern to me,” said Harren. “Those
walls are strong and thick.”
“But not so high as to keep out dragons. Dragons fly.”
“I built in stone,” said Harren. “Stone does not burn.”
To which Aegon said, “When the sun sets, your line shall end.”
As the last light of the sun faded, Black Harren’s men stared into the
gathering darkness, clutching their spears and crossbows. When no
dragon appeared, some may have thought that Aegon’s threats had been
hollow. But Aegon Targaryen took Balerion up high, through the clouds,
up and up until the dragon was no bigger than a fly upon the moon.
Only then did he descend, well inside the castle walls.
On wings as black as pitch, Balerion plunged through the night, and
when the great towers of Harrenhal appeared beneath him, the dragon
roared his fury and bathed them in black fire, shot through with
swirls of red.
Stone does not burn, Harren had boasted, but his castle was not made
of stone alone. Wood and wool, hemp and straw, bread and salted beef
and grain, all took fire.
Nor were Harren’s ironmen made of stone. Smoking, screaming, shrouded
in flames, they ran across the yards and tumbled from he wallwalks to
die upon the ground below.
And even stone will crack and melt if a fire is hot enough. The
riverlords outside the castle walls said later that the towers of
Harrenhal glowed red against the night, like five great candles … and
like candles, they began to twist and melt, as runnels of molten stone
ran down their sides.
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Conquest
So if Walls of Harrenhal can melt, so can the narrow passes and the walls of Eyrie.
