Take a clear picture of the part from the top and the side with a solid contrasting background and then use Google image search. Those look really old though. The plumbing supply house may have something that fits but which isn't the original part.
What is the problem with these? They look very similar to the internals for a hose bib, albeit bigger. So maybe a new rubber washer on the end and repack the cap and you're back in business?
Is there access to the other side of that wall? If so it might be easier to remove this and put on something which is still readily available. There doesn't seem to be any sort of caulk between the tub edge and the wall, and the wall is just painted boards. Is that a free standing tub?