Possible spoilers for AGOT:
When Lysa Arryn lets it be known that she will keep her armies in the Vale and not participate in the war, it is a blow to the northern/riverland forces. But how much of one? How many soldiers does the Vale have, and how formidable would they be in combat?
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.
So Spake Martin
In an e-mail George R.R. Martin outlines the strength of the Vale, and other domains, in comparison to Robb's forces:
[Interviewer] Quick question - We have seen all of the seven kingdoms in action in one way or another except Dorne and the Vale. I am trying to get an understanding of the various strengths of the different realms. When Robb calls the Northern Banners he gathers a host of about 18 thousand men. How do Dorne and the Vale compare to this (I don't expect numbers, just general feeling)
[GRRM] I'd say these three kingdoms were roughly equal in the force they could assemble... but the north is much bigger, so it takes longer for an army to gather. And life is harsher there as well, so lords and smallfolk both need to think carefully before beating those plowshares into swords.
[Interviewer] I also gather that Highgarden is the most populated place (Renly mentions 40,000 men), with the Lannisters being next (two combined hosts in GoT seem to be about 35,000).
[GRRM] The Reach is the second largest domain (after the north), and the most populous and fertile as well. The westerlands are the richest, with all the gold and silver in those hills. Those two regions also have the greatest strength at sea, after the Iron Islands.
[Interviewer] Dragonstone appears to be the least powerful. The Riverlands and Stormlords appear to fall in the middle I imagine.
[GRRM] Dragonstone is last, agreed. The riverlands are rich and fertile and populous, but suffer from divided leadership and a lack of natural boundaries. The stormlands have lots of trees and rocks and rain.
How Formidable Would They Be In Combat
In essence, The Vale of Arryn is a normal realm, with normal leaders1 and Knights2 and soldiers.
Their main asset though, would be their numbers; roughly 18,000 men. If added to the North's host, they would essentially double the size of Robb's army.
We saw what Robb, Roose, Edmure and the Blackfish, Rickard and The Great Jon did to the Lannister forces with merely 18,000 men.
Now imagine 36,000; that would be roughly 1,000 men more than the Lannisters. The North could take on the Lannister forces 1-to-1. This is why Robbs is so upset when he hears that his own auntie has declined to join the fight and help them.
1. Some notable Houses of the Vale:
2. Some notable Knights of the Vale:
