Tyrells were Targaryen loyalists till the bitter end. They dipped their banners only once King's Landing had been taken by the Rebels.
Tyrell debt of inherent loyalty to Targaryens
When the Dragons came with Aegon the Conqueror, King Mern IX of House Gardener ruled the Reach. Tyrells were only stewards of the Kings of the Reach. As it happened, King Mern chose to fight the invading Targaryens and perished in Field of Fire along with rest of his kin. The royal dynasty of the Reach went extinct in that battle.
When the news reached High Garden, it was the High Steward of the Reach named Harlan Tyrell who surrendered the Kingdom to King Aegon Targaryen. Aegon rewarded Harlan with titles of Lord Paramount of the Reach and Warden of the South even though other Houses of Reach had better claim on the titles due to their kinship to the extinct House Gardener. So, Tyrells got their status as Great House only due to Targaryens, unlike other great houses who were Kings once e.g. Starks, Lannisters and Arryns. They have always answered the call of the Iron throne and stood by the Targaryens in challenging times like Blackfyre rebellions, Dornish wars etc.
They however failed to play a much greater role for the loyalist Armies mostly because of the incompetence of Lord Tyrell and stubbornness of Stannis Baratheon. Following is a complete breakdown of their operations during the war:
Battle of Ashford
The Tyrells and Baratheons first clashed at the North-Eastern border town of the Reach named Ashford.
During the Rebellion, Stormlands were completely isolated from their allies in Vale, North and Riverlands. So Robert wanted to advance his forces north and link up with his allies before he could be trapped between the loyalist forces of Dorne, Reach and Crownlands. But he had to first deal with Storm Lords who were trying to rise up for the King.
Robert had won victories at Summerhall against those Lords and now had the complete support of Stormlands behind him. With home front secured, Robert advanced North-Westwards to link up with his allies; The Houses Tully, Arryn and Stark.
Naturally, to reach Riverlands, Robert had to either cross the Crownlands or the Reach. Robert chose to cross through the Reach and came upon the border town of Ashford. It is unclear if Robert tried to take Ashford castle before advancing to Riverlands but we know he was soon faced by Tyrells. Lord Mace had bestirred himself and Lord Randyll Tarly, known as finest soldier in the realm, had reached Ashford with his vanguard.
Tarly did not wait for main Tyrell Army to arrive and charged at the Stormlanders. Stormlanders suffered losses and on arrival of main Tyrell Army, Robert was forced to withdraw to Riverlands. This is the only defeat Robert ever suffered.
Since the Stormlander Army had retreated into Riverlands, the way was clear for Tyrells to lay siege to Storm's End itself, the ancestral seat of House Baratheon. That's exactly what they did. Lord Mace Tyrell advanced Eastwards and laid siege to the Castle of Storm's End.
Battle of Mander
This battle happened after the Battle of The Trident, where Prince Rhaegar died and the fate of the loyalists was sealed. Lord Paramount of Iron Islands, Quellon Greyjoy had so far been neutral in the civil war but in the aftermath of The Trident, he was persuaded by his sons to enter the war on Robert's side, lest they lose all chance to win glory and plunder.
Lord Quellon sailed to attack the Reach from rear with his sons. He took only a small part of Iron fleet with him because Lannisters were at that time still neutral and could pose a threat to Iron Islands if they declared for Targaryens. The Greyjoy navy plundered and burned their way along the coast until they were met by Tyrell naval units from Shield Islands. The Ironborn proved their reputation as greatest seafaring people in the Seven Kingdoms and emerged victorious. Lord Quellon Greyjoy was however slain and his son Balon decided to return to Pyke to take his father's seat. This was the only attack mounted by Rebels on Reach itself.
Siege of Storm's End
In the Aftermath of Battle of Ashford, Tyrells had laid siege to Storm's End. This made matters very grim for the Baratheons because their army was in Riverlands, cut off from their lands so no relief force could be sent to break the siege.
Tyrells had outstanding numerical superiority but they did not have one thing; the iron will possessed by Stannis Baratheon. Stannis refused to surrender the castle and kept defying the besieging Armies.
Since Storm's End is a very strong castle and it is said that the trebuchet which could punch a hole in its curtain wall had never been created, so Besiegers were happy to starve out the Baratheons inside the castle. Soon the Redwyne and Tyrell fleets arrived and Storm's End was cut off from sea as well.
Stannis, though out of food, refused to surrender still. He and the garrison instead ate cats, dogs and shoes but they held on, tying a huge part of Loyalist Army in siege of Storm's End which could have been put to other use elsewhere. If Storm's End had fallen, Tyrells would have joined the other Royalist Armies and the outcome of The Battle of The Trident could have been the death of Robert instead of death of Rhaegar.
But Storm's End did not fall, due to the iron leadership of Stannis Baratheon and supplies brought by a lowborn smuggler named Davos. In the meanwhile, Rebels killed Prince Rhaegar at The Trident and King's Landing itself fell to Rebels. Soon Lord Eddard moved with his men to Storm's End and notified the Tyrells about the outcome of the war. Tyrells, upon learning that Rhaegar was dead and Aerys was slain with most of the royal family, dipped their banners. The war was lost. Robert pardoned the Tyrells and their vassals and confirmed them in their rights and titles.