As one of the great houses, the Vale has a significant strength. It's never specifically stated how many swords they have, but there are a number of indications.
Game of Thrones
When Riverrun is besieged by Jaime Lannister, Bryndon Blackfish requests that Lysa send a thousand seasoned soldiers to their aid.
"I asked your sister for leave to take a thousand seasoned men and ride for Riverrun with all haste. Do you know what she told me? The Vale cannot spare a thousand swords, nor even one, Uncle, she said."
While Lysa says the Vale cannot spare them, this is most likely just her refusing to take part in any conflict, and keeping all her power close to the Eyrie. It's a safer bet to trust Bryndon, and proven warrior and smart man. So the Eyrie can afford to send a thousand seasoned men for one siege. This suggests that they have many more than that, and are a significant force.
Furthermore, Robb clearly thinks that the Vale could turn the tide in their favour. He is incredibly dismayed when he hears that Lysa won't be sending any troops in his aid.
“Is there word from the Eyrie? I wrote to Aunt Lysa, asking help. Has she called Lord Arryn’s banners, do you know? Will the knights of the Vale come join us?”
“Only one,” she said, “the best of them, my uncle... but Brynden Blackfish was a Tully first. My sister is not about to stir beyond her Bloody Gate.”
Robb took it hard. “Mother, what are we going to do? I brought this whole army together, eighteen thousand men, but I don’t... I’m not certain...” He looked to her, his eyes shining, the proud young lord melted away in an instant, and quick as that he was a child again, a fifteen-year-old boy looking to his mother for answers.
Robb has 18,000 men as his full strength of the North, and many of them won't be seasoned. If Bryndon was able to send 1000 seasoned men for the case of a single siege, then they will be able to muster up much more than that for a full war, significantly increasing the strength of Robb's (or anyone else's) army.