Do students at boarding schools ever get internships? If so, how common is it? And is it during the School year or over the summer?
Our son did a film internship the summer between freshman and sophomore year, but not through his BS. He found it on an internship website. We did not see him much that summer. 
I think you have to find your own internships. Lots of kids do summer programs at colleges which I would stay away from. They wreak of privilege and don’t help with elite college admissions. you’re better off trying to start your own business, find or do your own research or find a leadership position in your community volunteering and the like. A summer job if you can get it is also useful. If you volunteer, you can morph it into an unpaid internship.
Our school did have a connection with a local company who hired several interns to do coding over the summer, but I don’t know much about it beyond that.
DS got an internship one summer on his own. (More like work for free.)
I don’t think that most schools really emphasize this. Volunteering or a summer job is probably more realistic.
Several schools we visited did have internships and/or research opportunities. Also, some schools connected students to internships or research projects for Senior year. FWIW the girls schools we visited had such programs. There are several coed schools also with programs - some have research opportunities also leading up to Senior year that build upon Summer internships. This was important to us, so we applied to schools that offer these opportunities. There are schools with working relationships with nearby universities or medical centers, even though these may not be school-based internships. This was a consideration in where to apply. Feel free to PM for specific information.
2/4 had a published internship program. Both programs were in math/science. They were given to us by the AO when we visited. One tied into the curriculum ( pretty cool). The other looked like they had gone to alumni and gotten them to take an intern or two. Either way, they looked great for the right kid. In both cases, you had to apply and it looked like it was for the kids who were outstanding in math/science.
You can also always talk to the alumni office at your school, they are always willing to help you connect with alumni who could possibly help you get a internship
I think it really varies from school to school. It is mandatory at my daughter’s school for Juniors and Seniors to have internships over Jan Term. The school helps find some of the internships, while sometimes girls find their own. Jan Term takes place the three weeks after winter break, so often girls will find an internship in their hometown, but over the past two years, I’ve seen girls with internships in many different places and in many different fields.