Jane or any of our trained associates is/are available to assist you. I think both are correct depending on how many "associates" are meant. Also, note that you don't need to put comma after Jane.
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JRseaJane, or any of our trained associates, is available to assist you.This is correct because the subject (Jane) is singular.
tinanam0102What would say?If I paused before or and after associates, I would use is. Otherwise I would use are.