Once you're received the tap and done any necessary cleanup/renovation work on it to be serviceable you prepare your dowel1, thread a portion of it, then determine what the pitch and profile is. Once you know this you see if you can find a matching tap for sale somewhere.
Given the typical thread profile for modern wood taps and dies (90° V-threads, rather than the 60° typical of threads in metal) this might not be as difficult as it sounds, as long as the pitch isn't unusual.
The classic problem when seeking to start making threads in wood is that you could really do with having a die in order to make the tap, or the tap in order to make the die. Since you have one half of the equation you're well set up to be able to complete the operation.
In a nutshell what you can do is:
- thread some dowel;
- cut some relief for the scrapings2;
- shape and insert a cutting tooth3 into the leading edge of the thread, adjacent to the relief and close to one end.
Most users find putting a slight taper on the leading portion of a tap useful, although just a chamfered leading edge is also an option. The tooth is inserted past any taper or chamfer or it won't cut a full thread profile.
The cutter doesn't absolutely have to be tool steel, but while you can use mild steel I would suggest at least using fileable high-carbon steel for better performance (e.g. a scrap from a saw blade, any saw of any vintage will do). If you don't have an old saw or two lying around you can sacrifice a small portion from the tip of any unbacked saw without affecting its performance. If you're seeking to do this quite a bit, and/or in harder woods, it would be very beneficial to make the cutting tooth from a thicker chunk of tool steel, fully annealed for ease of working. Allen keys are usually a decent source of suitable steel. After shaping you harden and temper it, before final honing. Alternatively, use the stub of a broken HSS drill bit which won't require any heat treating, but must be worked in a fully hard state so grinding or diamond files are generally required.
1 This isn't just turning the wood to the major diameter, in case you haven't seen this tip while researching threading wood it's apparently very beneficial to pre-soak a dowel to be threaded in oil prior to cutting the threads. The most common choice for this seems to be linseed oil but some use mineral oil and get comparable results, so I suspect the type of oil doesn't actually matter.
2 Since it's easiest to make and insert just one cutter you'll be making a single-flute design.
3 Obviously you shape the tooth as accurately as you can, but wood threads of this kind are not high-precision tooling so just hand-filing or grinding a tooth to approximately the right shape can still yield a thread profile that works, and works well.