All you need is a valid Equity contract (Showcases, 99-seat agreements do not count) to be eligible. This sounds like a valid contract.
Once you've signed it, you can do one of two things:
1) BRING the contract to Equity. You'll have to pay a minimum of $400 of the $1100 initiation fee. You'll fill out an application and be issued a temporary card then and there.
2) WAIT for the theatre to send Equity the contract. Once received, you will be sent an application to fill out via mail.
I would recommend the former (or mailing your contract yourself).
Depending on how long your contract is, Equity may request deductions from your pay to go toward the initiation fee. You are NOT a member until that $400 of your initiation is paid.
Don't follow goldenboy's advice to pay your employer back so they offer you a valid contract. That's just not sound advice.
"...everyone finally shut up, and the audience could enjoy the beginning of the Anatevka Pogram in peace."