I’ve rooted tomato suckers several times and it is quite easy. I usually do it if I see blight setting in and I’m worried about losing a particular variety for the season. I find a sucker that still looks free of blight - suckers between 6-9 inches long work well - trim off any lower growth so that all leaves are off the bottom 1/2 of the stem (sometimes no trimming of leaves is required) and put them in very moist potting soil with the bottom half in the soil. Then I put that pot inside or in heavy shade and just wait. Most look wilty for a few days, but perk up. If it is really hot out keeping them inside might be better, but be careful if your humidity is low inside from the AC. If keeping them inside I put a loose plastic bag over the pot and that helps keep them from ever going too wilty. I pinch out any buds on the sucker before rooting and any new buds until the plant gets some size. This is much quicker than starting new plants from seed and they seem to get into fruiting mode much faster than a seedling as well.
I expect peppers would be similar, but might take longer to root so I would use a plastic bag loosely over it or a the cut off top of a clear soda bottle and probably use rooting hormone to get it going faster.
If that fails, I would try doing a semi-layering by putting a little rooting hormone on the base of the branch I wanted to root and just wrapping a little wet sphagnum moss around that in some saran wrap. I’d check it every few days and once I saw roots starting go ahead and cut it off and put it in the moist potting soil with the humidity cover. I don’t think you’d need much moss and probably don’t need to wait until you see real long roots - I think it would work as soon as you saw the roots just starting. But since I haven’t tried it with peppers this is just what I think would work.
Let us know how it goes.