Is it possible to fit regular sunglasses with prescription lenses at the optometrist?

Sometimes. There are difference between inexpensive "fashion" frames and "ophthalmic" frames designed for custom lenses. Visit your optician and see if your frame can be pattern matched for a prescription lens.
Yes. Optical shops commonly incorporate your prescriptive needs into the sunglass so that you do not have to wear something like a clip on sun protection.