I saw this earlier today and just noticed it hasn't been responded to so I'll touch on some of the basics.The stem needs to be a hardened steel so it can sustain repeated impacts from the hammer without mushrooming or bending. O1 tool steel is a common material. It is sold in an annealed (soft) state and can be hardened at relatively low temperatures and quenched in oil (the O in O1 refers to oil). A Google search will turn up lots of good DIY guides on hardening O1 for things like making knives. Or if that's more than you want to deal with, you can buy hardened rod and cut it to length and press it into the poppet with a slight interference fit. The disadvantage is that the hammer might eventually drive it through the poppet, breaking it. Whereas if you make a stem from an annealed tool steel, you can thread it (and the poppet) which will make it more durable. That's not to say the former is destined to fail, just that the latter will be more forgiving.The poppet is typically made from a plastic of some sort, usually either acetal / Delrin or PEEK. Delrin is a good all-around material on a balance of cost, workability, and performance. PEEK is a harder material which has the advantage of needing less hammer strike to knock it open but the disadvantage the sealing surfaces must be nigh on perfect or it will leak. That is, Delrin is ever so slightly compressible so it will better conform to the valve seat and seal properly if the surfaces are less than perfect.Like most things, how you approach making a valve stem assembly will depend on your skills and tools. It's easiest to make one with a lathe but it's also very much possible to do with hand tools and a dose of patience.With a lathe, turn a cylinder in your preferred material. Center drill it to roughly 2/3 depth to accept the stem. For example, for a 0.35" diameter poppet that's 0.5" long, I would drill it about 0.3" deep. Then either press in or thread in the stem as described above.If you don't have access to a lathe, you can do it with a drill press or a handheld drill. Cut a piece of the plastic rod stock to length and center drill it to the proper depth and press in a piece of hardened rod. Then chuck up the rod in your drill or drill press and get it spinning, and use a burr in a Dremel to reduce the plastic rod down to the diameter you want. Then you can smooth the surface with a file and/or sandpaper. Now carefully face off the sealing surface as best you can. Don't concern yourself with getting it perfect at this point. The idea is just to remove any high spots. Don't use a file for this step because if you attempt to hold a file against the surface, it will just ride up and down over the high spots and low spots. Instead, use a burr in a Dremel and slowly bring it toward the surface until it just kisses the high spots as the poppet spins around. Eventually you'll get to a point where the burr is taking a bite for the full rotation of the poppet. You can stop when that happens.Now do the final truing with a piece of sandpaper held against a flat backer. Here's a lapping jig I made with a scrap of wood. I drilled a hole just big enough for the stem to pass through. Then I used a hollow punch to knock out a bunch of holes in a piece of 220 grit sandpaper. From there, it's just a matter of putting the stem through the sandpaper and the wood block, and grabbing the stem from the back side with a drill and spinning it against the sandpaper to smooth it out. If you color the face of the poppet with a marker, you can monitor for the low spots as you go. It will be obvious when you've done enough because the marker will be completely removed.Lastly, it's a good idea to lightly lap the poppet to the valve to ensure they mate fully. Again, color the surfaces with a marker and spin them together with a mild abrasive like J-B bore compound or a metal polish. When both surfaces develop a neatly polished ring where they meet, you're done.I hope this helps!
You can still use your 3/8 chuck as Jason said center drill, press in a shaft and chuck that in the drill.Amazon has PEEK on sale right now, 45% off.