There are a few related tags, e.g. shop=trailer, rental=yes, self_service=only. From what I can see no combination of the above really describes the situation accurately:
For a business exclusively renting out cars the tag is amenity=car_rental. It can be used without shop=*. Following this tagging your selv_service trailer rental would tag as
amenity=trailer_rental (undocumented but used some 25 times)
outdoor=yes
self_service=only
followed by tags for payment methods, brand, operator etc. etc.
I haven’t tried it myself, but my guess is that it has an internet connection and it’s unlocked remotely. The website says “The [booking] link is your digital key!”
It sounds a very niche business as it relies on having a towbar, with associated electrics fitted to the vehicle. Something only a very small number of vehicles have. Only those who already own a trailer or caravan are likely to have had a towbar fitted.