I don't have experience with these companies, but my hunch is that while it'll be hard for a high school junior to get a position, it doesn't hurt to try. Just don't put all your eggs in this basket. There's so much competition amongst college students to get these jobs, and since they've got higher education and maybe more specialized skills, they're at an advantage. You're also going to have a bit of a problem if you can't go up for in-person interviews. I know that Roundabout has been interviewing people this week, so you'll have to figure out how to get to NY ASAP!
But don't let me discourage you - one of my biggest regrets from high school and the beginning of college was not interning enough in a field that I'm interested in. It's going to be difficult in your position right now, but if you send out a lot of CVs (and keep in mind that smaller theatres will be looking for people too - don't limit yourself to the big guns), you may get your shot!
"It's not always about you!!!" (But if you think I'm referring to you anyway, then I probably am.)
"Good luck returning my ass!" - Wilhemina Slater
"This is my breakfast, lunch and f***ing dinner right here. I'm not even f***in' joking." - Colin Farrell