Sorry the PLA weren't much help. Any joy with Sure Start?
A few other ideas, in no particular order:
When you say you're Vice Chair, what's the Chair doing in this scenario?
Do you have Supervision & Support sessions lined up with the preschool leader / supervisor? Ideally you need to be sitting down together on a regular basis to review what's happening across the preschool, exchange ideas, discuss her training needs etc. She is responsible for managing the rest of the staff, but the Chair is responsible for managing her.
Are there particular training needs that she has that will help her manage the staff better, or do the staff have training needs to help them do their jobs better / without the conflict? If so, get it booked. Best case, it helps everyone. Worst case, you have at least got a record that you approached the issue constructively.
Enlist the help of other committee members. Chair is a lonely position if you don't get some moral support from other people, or even get another committee member to sit in on meetings with the staff.
You could call a meeting of the staff and lay it on the line that the issues between them are having a negative effect on the atmosphere for children & parents, and that it needs to improve. You don't have to be really nasty - ask them what they perceive the situation to be, and how they think it could be improved.
Define what you & your committee members are required to do. If you want committee job descriptions, there's plenty of people on here who can help you with that.
You say that you have let the staff tell you what to do as they have been there a long time. If there's specific issues / questions there, then again you can raise them on here to get an alternative view. You might get some helpful suggestions from other settings that you can take to your staff. People being in a job a long time can be brilliant in terms of experience and knowledge, but there's a risk (in any job) of getting stuck in a rut.
It sounds from your post that you do want to make a difference, and if you're more confident about what your role entails, you will be able to move mountains!