Is it possible to hook a portable clothes washer up to the hook-ups for a full washer instead of the faucet?

Yes you can... but the hose attachment is where you encounter problems: it is designed to fit on your kitchen sink spout. You may have to modify the hoses or replace them completely. Drain hose is no problem and requires no modifications. Just insert it into drain opening.

Milo did not go deeply into that problem. You have one incoming fill hose currently. You need to make a manifold so that line splits into two line reaching bot hot and cold. Most all of the stuff you need is right off the shelf.