My husbands ex-wife left the state without any written permission from him.
She said she would be gone 1 school year and now we are on the 2nd school year.

They have joint custody.
Can she stay away with them?

This depends on the way the custody. Specifically the visitation, was ordered by the court.

If there was a visitation schedule in place and her moved prevented the visitation schedule from being enforced, then she had to get approval from the court to move. If she didn't your husband needed to file a contempt of court to have her return. It may be too late to do so now.

But without knowing what the visitation schedule says its hard to advise.

What state are you in? As noted it might be too late now because he should have filed contempt charges when she first moved (providing he had a visitation order). The visitation is the only thing that matters because there are two types of custody physical and legal. If he was supposed to have joint physical custody (rare) then he should have the kids half of the year like all summer plus weekends during school. However, if he did not have that or any other visitation then it really dosen't matter.

As noted, it will depend on their custody agreement, what does it say, but it appears since they are already gone, has been for a year, even if there was clause in there, I doubt it could be enforced any longer