Many optical stores will do this for you. The lab technician can shape the lens to fit your old frame. But it may cost you more than a new pair of prescription reading glasses because replacing the lenses with prescription ones requires a lot of work. If I were you, I would buy a new pair of prescription reading glasses.
I think you are meaning you want to keep the frame of your reading glasses and get your prescription lenses in it. Yes, you can. Many optical store can cut prescription lenses for the frame that customers offer. You can go to Target and ask.
If you want to save money, I suggest you to buy a pair of new glasses online. Usually the same glasses sold online are much cheaper than that sold in the physical shops. I have a pair of glasses which has been used for more than 3 years. Now I want to buy a pair of new glasses too. So I took my old glasses to an optician. He told me that I'd better buy a pair of new frames for the old frame was out of shape. So I doubt whether your frame has the same problem. Hope this is useful!