One of the requirements for a loan to be subject to Reg Z is:

(iii) The credit is subject to a finance charge or is payable by a written agreement in more than four installments;

Does this mean any charge that fits into Reg Z's definition of finance charge, or does it mean the borrower has to pay interest? I came across a loan (for down payment assistance) that charges no interest, and has no monthly payments (just a single payment), but it was charged an origination fee. Will the origination charge fit the requirement above?

Yes.