I would allow Moon Druids to do it (frankly, I'm a bit surprised they can't.) If the character isn't Moon, but wants to become an owlbear anyway,* I would work with them to figure out something special to their character. Because the existing owlbear is probably a bit too powerful for non-Moon Druids, but I could see a specialized owlbear variant working that enhances the stuff the character does do. (E.g. I could see a fiery "embear" version as an alternative to summoning your fire spirit for Wildfire, or an "astral owlbear" as an augmentation of or alternative choice within the Starry Form option for Stars.) This would, naturally (heh), be something the character must build up toward, or else pay for by giving up something else in its stead.
*As noted by at least one and probably several prior posters, that's a thing WoW Druids could do, "moonkin" form is the caster specialty for their Druids. "Moonkin" is the Blizzard-proprietary name for owlbears in their universe. So there's literally millions of people out there who are used to associating owlbears with druid transformations.