How to obtain carpet

Try a local carpet store.

Ditto the local carpet store advice. They can cut wall to wall to the size you need and bind the edges with stitching. You could then put it down as is if you don't mind the harder feel. Or put padding down underneath it. They may also have some carpeting with a slip resistant backing suited to what you are trying to do.

The internet have vast amounts of wonderful information. plenty of catalogs of sellers, but . . . . . .

Sometimes it is best to take your ass out of the house and down tot he local store. Most towns have at least one carpet dealer that can do exactly what you want. Go visit one.

Ask for "low loop commercial".

We investigated purchasing carpet from Home Depot and having it bound - they will do it, but use an outside company to do the binding. There is "indoor/outdoor" or kitchen carpet that would suit your needs - thin, short nap. Just visit a good carpet store. There is probably some sort of coating that can be applied to the back of carpet if not already part of it. There are cut to fit "pads" available in most home stores - just a mesh that is coated and folds up. The leather or vinyl might be a problem - cracking or creasing when you roll it up to move it. Whatever backing you consider, be sure it won't stick to the floor underneath. If the large area rug fits up against the walls, you may not need any backing - not likely it would slide, but may curl if not fastened down.

Local store - I have bought from S&S mills of GA. They will send you free samples, you order and they ship to you. Not sure if they can do the binding. Doesn't hurt to ask.