The day of your third missed Form 990 (i.e. May 15 on year 3), the IRS will automatically revoke your nonprofit status. THIS IS BAD.
Now, you are operating as a for-profit company in the eyes of state and Federal law. That may change a bunch of stuff for you.
Generally your state's laws require that you have a bunch of stuff in your Articles of Incorporation or Bylaws that assets must be dispositioned to other nonprofits when you disband. Losing your nonprofit status places you in the weird and serious position of those assets coming over from the nonprofit space into the profit space, which could create serious tax issues. And the liability for those may fall through to the responsible managers of the nonprofit.
This is not an outcome you want. Since filing a Form 990 is free, I suggest you keep doing it!
It's extremely rare for the government to require a credit card. Generally they take credit cards begrudgingly, and only because it's a practical necessity for allowing online transactions, and often only through a distant third party. It's not their preferred M.O.
If you think the government requires a credit card, you are thinking only of the online-transaction option. The normal, default way to handle registrations and payments to the government is by check or money order sent postal mail. You should look into that method, and try it.