It can be done, and it has been done .... to a high professional level. It's rare. Even if the student doesn't have a regular teacher, the ones who do well without are usually in contact with other players and pick up things from them.
Other instruments are commonly self-taught to the higher levels .... guitar comes to mind. I know a whole slew of excellent professionals who never had a lesson on guitar, and many of the biggest names didn't either. Bass, drums, vocals and a bunch of other instruments can be very successfully self-taught. Trumpet is not one of those. Lessons are pretty much a must.