When our helpful gov't (
) changed the 403b rules and regs a few years ago, they included some employer responsibilities and other provisions that complicated the game. Employers generally responded by dealing only with vendors that would handle these things for them. The services, of course, wind up being paid for by the employees through fees. Not all financial houses wanted to do this. Vanguard, for example, dropped out at DW's pre-RE employer.
When OP called Vanguard to transfer his account, that indicated he hadn't done his homework. Knowing which providers your employer has contracts with would be an extremely basic beginning. It sounds like that info is understood now. The next step is to contact the Fidelity representative assigned to his employer. The employer (if a school district that would be the business manager) will have the name and number. That person will know what to do and can get the job done. Fidelity and other providers do not want phone reps to handle 403b issues, again due to the complications of gov't regs now in effect.
DIYers often find this confusing/frustrating. But it is the way it is. Even with Fidelity, you will be assigned an FA. Calling in on a Sunday afternoon isn't likely to get the job done.
Edit: The above is based on my experience helping DW xfer her 401b several years ago. No guarantees that things are exactly the same today.