How different would things have gone if the Vale had declared for Robb?

[In A Coat of Gold]

Even if Stannis takes KL, his power will still be crippled enough that he won't have to worry to much about him. Simply, siding with Tywin when he is completely surrounded by the North, Riverlands, and Vale is just as risky. Especially, as it is likely that Robb would have already attacked and broken Tywin's personal forces if he had the additional boast of the Vale. Meaning if anyone it would be Robb he would reach out to.

Edit: Moreover, all it takes for Mace to be somewhat delayed in his assistance and the Lannisters falter at BW.

Siding with an experienced miltary commander who still has the majority of his host, a strong castle, and the heir to the throne as his grandson isn't that risky. Everyone knows about the Tyrell's massive host, Tarly, the Redwyne fleet.

If Tywin marches on Harrenhal, the mountain clans still killed the scouts. He would still take them unaware and assault from the rear.

[James Steller]

Siding with an experienced miltary commander who still has the majority of his host, a strong castle, and the heir to the throne as his grandson isn't that risky. Everyone knows about the Tyrell's massive host, Tarly, the Redwyne fleet.

If Tywin marches on Harrenhal, the mountain clans still killed the scouts. He would still take them unaware and assault from the rear.

Not necessarily, as Tyrion might not have been released from the Eyrie to make such an alliance.

[In A Coat of Gold]

Not necessarily, as Tyrion might not have been released from the Eyrie to make such an alliance.

They don't need the mountain clans specifically. If Tywin sends Kevan, the results are most likely the same..except the City Watch has a seasoned knight leading them to battle rather than a dwarf.

[Rohirrim]

IMO the Vale declaring for Robb would have prolonged the war. In regards to Stannis, he is an unbendable character throughout the series especially early on. If Robb's mission was for revenge on the Lannister's he could have worked with Stannis' and the combination of the Vale, Stannis, and the North would be more than the Lannister's and the Tyrell's could overcome. Even after Stannis' defeat at Blackwater, the battle weakened the capital significantly. I would say that all that Robb needed was to defeat the armies everywhere else and the capital would have likely fallen to him. Declaring himself King of the North brought him the disdain of Stannis, because if Robb and Stannis' succeeded then Stannis lost a significant amount of his future kingdom. As a king, Stannis wouldn't simply eat his crow and take the rest of Westeros as a conciliation prize. Robb screwed himself by declaring him the king of the north will no clear cut plan after he sacked King's Landing and killed the Lannister's. With no claim to the throne he couldn't rule the whole Kingdom for long, and giving the crown to Stannis would have been enabling a future conflict when Stannis wanted the North back. If the Vale declared for Robb, and Robb simply wanted to remain Warden of the North and support Stannis as King then I don't see much of way for the Tyrells and Lannister's to win.

A melee war for 5, 4, or even 3 armies is not sustainable. As time progresses the number of factions gets reduced as new alliances are made based on the realities of gains and losses.

There's still the matter of the Iron Born. They would have been a thorn in Robb's side throughout the conflict. Regardless of if Stannis/Stark/Vale were in alliance or any of the other combination, if Tywin was losing to Robb he would have eventually reached out to Balon. Balon maybe could have declared himself King of the Iron Islands and seceded amidst the conflict while holding out on his Islands. Declaring himself King of the North was too optimistic IMO. As mentioned, the Iron Born's strength is more likely in their naval fighting and raiding versus the ability to hold castles and lands. Stannis, Tywin, nor Robb would have allowed Balon to hold the North and we would have lacked support from the local lords as well. Nor would Stannis or Tywin allow the Iron Islands to secede. I think inevitably as the war prolonged more in Rob's favor regardless of who declared for him, Tywin would have reached out to Balon and let remain Lord of the Iron Islands, and a new title of Warden of the North as a conciliation prize for his support. Despite Balon's posturing he didn't uphold the Iron Price that he played so much lip service too and would have probably taken the deal.

On a side note, the longer the war plagued on the weaker the kingdom would have been as a whole and Dorne probably have declared themselves free again given their already thin ties.

[Minsc]

Siding with an experienced miltary commander who still has the majority of his host, a strong castle, and the heir to the throne as his grandson isn't that risky. Everyone knows about the Tyrell's massive host, Tarly, the Redwyne fleet.

If Tywin marches on Harrenhal, the mountain clans still killed the scouts. He would still take them unaware and assault from the rear.

Tywin wouldn't likely have all of that if the Vale joined. As Robb would have likely broken his host even more then he did in the books.

Similarly, Harrenhal would likely be surrounded thus not making it a secure base.

Just because in one case they join the Lannisters doesn't mean they will do it when they are even weaker.

Moreover, as I mentioned all it takes is their decision taking a little longer and they are screwed.

Furthermore, they didn't act until Tyrion reached out to them meaning that if Tyrion's fate changes because of the Tyrell alliance is hindered