If the plastic is thin, it probably does need reinforcement. You could take it in, but if you're a do it yourselfer, here's a way:
When you get the strap button and screw, find the proper drill size for the screw, so it will thread into a piece of wood without too much difficulty, but with enough resistance that you know it's a good strong connection. Take a small piece of wood (plywood won't split) about an inch square or round, and drill the proper size hole in the middle of it. Thread the screw into the hole so the threads are already cut and you won't have to bear down too hard when tightening it on the uke. Drill the uke in the center for the screw, measure for the center and mark the hole so you get it right. You may want to loosen the uke strings to make things easier from this point.
Now feed a string...an old uke string, fishing line, thin flexible wire, or something through that new hole in the uke, and grab it through the soundhole and pull it out a few inches. Run the string through the center hole of your drilled wood piece, and tie it to a washer, nut, or something too big to fit back through the hole, so when you pull on the string, it will pull the wood block into the uke and into place.
Put a little glue...Duco cement, weatherstrip adhesive, or maybe even superglue on the side of the wood that will go against the top of the uke when it's inside. It really doesn't have to be all that strong, but needs to hold it in place while you install the screw and bear down on the screwdriver. Lay the uke on a table, pull the wood piece snugly into place, and hang something that weighs a couple of pounds on the string so it keeps the wood tightly in place while the glue dries. When it's dry, cut the string, turn the uke upside down and shake the washer and string out the sound hole, thread in your button screw, and you're in business.
Note; instead of the glue, you can use a couple of pieces of double-sided adhesive strips, such as the 3M Command mounting strips sold in hardware stores, used for hanging pictures etc. They are very strongly adhesive, won't have any issues with sticking to clean wood or plastic, and may be less messy than glue...plus no waiting to dry.