Neither appears to be documented on OSM wiki or in presets.
shop=beekeeping_supplies seems best one as it is clearly about supplies for beekeepers, not about honey or something produced by beekepers
Note that I am not planning to make proposal or even create a documentation page as I lack knowledge here - but if you are familiar with this topic then documenting one of in use tags is going to make highly likely that it will end being a standard, and it is a good moment to choose something suitable. Or maybe alternative tagging like shop=trade trade=beekeeping_supplies is preferable? Then it is still viable to ask all people who mapped such shops and discuss it with them.
BTW, a half-century ago I kept bees with my dad (who taught me), who kept bees with his dad (who taught him, in Michigan), who kept bees with his dad (who taught him, in Hungary)…I can trace my beekeeping back to the (late) 19th century.
There are such shops, they are sometimes “rolled into” other (craft-style, DIY) shops. There is a gardening store near me that dabbles a bit in apiary supplies. You might also find such supplies at big-box stores (or not), you might find that a certain amount of this has gone to on-line commerce (but not all of it has, of course).
I also like shop=Beekeeping_Supplies, though I think I prefer shop=beekeeping_supplies (without the CamelCase, lowercased). It is about supplies, not necessarily finding another apiary. Well, thoughtfully, I recall years ago the gardening supply store had a few good soft-bound “general purpose” books on beekeeping that weren’t too bad for a beginner wanting to get started, too.
It would be good to have a summary of usage of each of these tags as well (a crude one in taginfo), but I agree beekeeping_supplies is the clearest of these used, but shop=beekeeping seems most popular. I’m not sure if there is ambiguity associated with the latter.
I wasn’t aware of one mapped locally, but turns out there is one. Given the population catchment is about 1% of UK this suggests there may be 50-100 countrywide.