Im not sure anymore about annullments, my diocese called and said they cannot process it, despite having witnesses write in, because I havent paid any money toward it. Said I cannot, trying to support myself and my child alone. Beginning to think its all about the money.
Did whoever you spoke with actually say "the tribunal would not process your case because of non-payment?" If so, that employee had their facts wrong and you should, as Laura suggested, call back and speak with someone else.
Technically, there is no charge for the annulment process. Tribunals ask for a certain fee to be paid in order to recoup their administrative costs but not being able to pay in no way should affect the processing of your case.
I don't know of any tribunals that will not take a payment plan. In my own case, since I was eating hot dogs and Top Ramen for a long time, the Tribunal secretary let me make a $10 a month payment, but not before telling me that if I absolutely could not afford it, I didn't have to pay anything. I strongly encourage you to call your tribunal and speak with someone besides the one you spoke with, or at least get more clarification. If whomever you speak with insists they won't process your case until you pay, then I would promptly write a letter to your bishop, explaining your situation.
Financial hardship is one of the worst things to go through! Sorry you're in that position. Count on my prayers for you.