What if Robert Arryn had been fostered at Dragonstone with Stannis Baratheon? Would the Vale have rallied to Stannis’ cause when crowned himself? Could the Vale have mobilized enough troops to make a difference?

Yes and yes. 

As Robert’s foster father, Stannis would almost certainly become the Regent of the Vale rather than Lysa Arryn. 

As Regent, Stannis would command the Vale’s army of some 30,000 and their fleet at Gulltown – this would allow him to go for King’s Landing right away, rather than diverting to Storm’s End. Without the time to prepare, Tyrion would have not been able to stop him from taking the city and Stannis would have sat the Iron Throne. How long he would have kept it is another question.