Yes, you can use a jam chuck. You can also make your own chuck from a block of wood and a carriage bolt- either designed to mount in a scroll chuck, or that is threaded to fit directly onto your lathe spindle.
Whether you need a tap depends on the style stopper you are making. Metal stoppers as sold by CSUSA, PSI, Ruth Niles, etc, require a threaded hole to mount the stopper on the turning. But if you are using either the simple cork design, or the ribbed silicone design, the stopper mounts by gluing a dowel into the turning. In that case, you don't need a threaded hole.
You can thread a hole in soft wood using a 3/8" carriage bolt. But that probably won't work in a harder wood or in plastic.
I finish my stoppers using either WOP or waterbourn poly. Either takes several hours to apply enough coats while allowing drying time between applications, but if you make your own mandrels, you can work several stoppers in parallel. WOP has a slight amber color that looks very nice on wood with a strong grain, while the waterbourn version is essentially colorless and doesn't change the color of the wood. I usually apply 6-10 coats to build up a nice finish.
The other trick I use is to bore a 7/8" hole in the blank, about 1/16" deep, and then bore the mounting hole in the center. The shallow hole allows the stopper to recess slightly into the turning, so that the join between the turning and stopper is hidden from view when when the stopper is placed on a bottle. Looks a bit nicer.
