Lysa, as a foreigner and a female leader, and an unstable one at that, seems to engender an unusual amount of loyalty from the lords of the Vale in refusing to enter the war of the five kings. Is this a result of Littlefinger bribing lords or something?

We get a pretty straight-forward answer to this: 

“Others believe that Lysa must marry again, and soon. Already the suitors gather like crows on a battlefield. The Eyrie is full of them.“

“I might have expected that,” Catelyn said. Small wonder there; Lysa was still young, and the kingdom of Mountain and Vale made a handsome wedding gift. “Will Lysa take another husband?”

“She says yes, provided she finds a man who suits her,” Brynden Tully said, “but she has already rejected Lord Nestor and a dozen other suitable men. She swears that this time she will choose her lord husband…it seems to me Lysa is only playing at courtship. She enjoys the sport, but I believe your sister intends to rule herself until her boy is old enough to be Lord of the Eyrie in truth as well as name.”

The Lords of the Vale follow Lysa Arryn up until she marries Littlefinger, because many of them were hoping to win the competition for Lysa’s hand, and through her to win the Regency of the Vale.