That's tough. Typically, I've utilized flats on wheels (with canvas covering) when no fly system has been available. I don't know any fabric you could paint and then scrunch without the paint cracking and th fabric wrinkling. One thing that works well is having flats attached as triangles, with one scene on each side (and then you push them together, rotating for each scene). You can also work with minimal backdrops and focus more on stage props and free-standing ply-board cut-outs to add dimension and set the stage. I have also used flats that were NOT on wheels, but were able to slide out and over an existing backdrop (like patio doors). As for matierials, I only have experience with canvas, and regular house paint works well, but be sure to tell the helpful person at the paint store what you're doing, and they can point out the best stuff to use (some is more glossy, others more matte). Good luck!

I am in a similar situtaion, with no fly system. Our backdrop is made of canvas...rather heavy. We did just paint ours and used regualr paint. If that is your only option (pulling it closed) a canvas shouln't be too bad for the first time or two you paint it, after that the paint can start to chip off. That bedsheet idea is sounding pretty good too, might be woth a test on a smaller sheet. Good luck!!

We used muslin, not canvas. Lighter than canvas, muslin comes in various widths so there may not be a need to sew pieces together. (Look up Rose Brand.) Seams are rather unseemly. Because we had no fly space, we rolled the drops up. Rollers are made with long strips of 1x1 inch wood nailed over six or so depending on length) 4" disks cut out of plywood. A system of clothesline and pulleys rolls the drop up and away. In this system, the drop is attached on a pipe at the top and the roller is resting on the floor. When pulle, the roller rolls up and takes the drop with it. It's hard to describe the setup with words only. Look up a good book on making scenery. As for paint, you may use house paint. Actually, we use the cheapest paint. It's not expected to last forever, anyway. Depending on the subject, we may water the paint down to keep the thickness to the minimum. You'll be surprised how quickly the weight of the paint adds up.
The drop will last through several paint-overs. We had one that lasted through twenty shows. We finally had to dispose of it when it became too heavy for three men to move. If you do the math, every time we painted it, it took about a gallon of paint. Twenty shows = twenty gallons of paint. Try moving twenty gallons of paint at once!