Of course!!! You can go into whatever group you want to join, and don't even have to own that type of animal. 4-H wants to teach the kids about animals and responsibility. I never showed horses, but I was in the horse group for 4 years This year I was in a goat/lamb combination group and learned plenty of good information about lambs as well.
4H is so much more than showing! There are all types of projects to do- electriciry, wildlife, robotics, entomology, healthy lifestyles, cars, small engines, presentations, officer positions, trips and so much more! Showing livestock is just one of the wonderful things you can do in 4H. Just about anything counts as 4H projects.
Reading what you wrote.. of course he can join without showing. Since he doesn't want to show which is a big part of it he might feel/be a little left out when it comes closer to show season I believe. When my 4-H club gets closer to show season and fair our meetings consist mostly of talking about showing & fair.
Check with your owns 4H extension. They are all so different. Some say you have to show in a certain number of shows to complete the year others do not make you show at all. Now that is is an animals project, like others have said there is a lot more to 4H then showing livestock.