I'm owed money from a non-profit for services rendered, but they are unresponsive

If the organization is going under, as an independent contractor, you're in the back of the line with all the other unsecured creditors. As opposed to employees who are at the front of the line. In most bankruptcies, that means that you will get pennies on the dollar of what you are owed if you get anything at all. And be aware that if the organization files for bankruptcy shortly after paying you, the bankruptcy trustee can (and will) claw back any funds the organization sent you. This can be a rather nasty surprise for contractors that have to pay back several months of invoices when the company goes under if they don't have that cash sitting around.

If the organization is solvent but simply choosing not to honor their debts, you could certainly sue them in small claims court. It seems unlikely that a reputable organization would suddenly decide to start stiffing a contractor that had been working for them for 2 decades on a relatively small invoice, though, so it seems unlikely that this would be a terribly fruitful path.

I'd continue reaching out to any contacts you have at the organization (presumably you have several after 20 years) to see if they can shed some light on the situation. You can send past due notices to the organization and, if your contract allows it, start assessing late fees.

It is possible that you'll just need to write off the invoice as an uncollectable debt. If the organization depends on government grants, for example, and those grants are held up at the moment, the organization may be insolvent at present. If there is no money to send out, you can't force money into their pockets so that they can pay you. Unfortunately, that's one of the realities of life as an independent contractor.