What kind of paper can I use in place of U.S. currency for origami?
I don't think any such thing exists; the two closest alternatives I can think of are both long out of manufacture (Eaton's Corrasable bond, and Crane's "Old Money"). Old Money would also have been too thick, but Crane took advantage of being the manufacturer of US currency paper by creating a 100% cotton bond paper that was 30% recycled currency. (Similarly, their "Denim Blues" recycled cotton scraps they got from Levi's, and is also discontinued).

Tracing paper is thin and strong enough, but holds creases too well and makes it very difficult to reverse them cleanly. Baking parchment (which you can find in food storage sections of stores, next to the aluminum foil and plastic wrap) has a similar issue, but is a little easier to work with, and in fact one youtube video demonstrates making a flasher with it.

The best alternatives I've found for ready-made papers to fold with that are thin and strong and take a crease well and of a size to allow for more complex models are the usual hard-to-source candidates, such as thin kraft paper. The origami-shop.com website has a good (expensive) selection of the papers folders prize in sizes larger than the usual 6"x6" kami that frustrates us so.

There is, of course, the option of DIY tissue foil (i.e., laminating foil with tissue paper and spray adhesive), and sizing paper with methyl cellulose but those never appealed to me, either.

See also: What should I look for in a paper for a complex origami model?