What an utter jerk. First, do as PP suggested and print or email to yourself any evidence you have of how long you've worked there.
Then go over to your boss' desk, or into her office or whatever, and say "excuse me X, sorry to bother you, but that background check form has to be completed today, can you please do it now, I'll stay here while you do it and give you the information to speed up the process". And DO NOT LEAVE until she does it. If she says she'll do it later, say that you are happy to wait and just keep standing there next to her. If other employees can hear you, so much the better.
If she says that she is not going to do it, ask her to forward the link to the women who does HR so she can do it on her behalf. If she refuses to do that to, look her in the eye and say "Why? Are you really the sort of person who would try to destroy someone else's chances of getting a job she wants?"
I know that's super scary, but you have to stand up for yourself. If none of that works, call the new employer and explain that they sent the forms to the company owner who won't fill them in as she doesn't want you to leave, but you can provide other evidence of employment and/or they can talk to the HR person.