No diocese makes money a requirement for an annulment. Dioceses have suggested payments as a way to help them defer their costs of processing annulments (i.e. salaries, mailing, stationary, phone, etc.). However I have never heard/seen anyone turned away because they lacked the money to pay the suggested fees.
As for the perception that annulments are expensive, it is erroneous. In my diocese the suggested payment is $500 which is split into 3 payments at different points of the process. Most dioceses are around that number depending on the costs of the area. But once again, not being able to pay the entire amount or anything at all does not impact the process.
