There are, and I'm sure you can apply whether you're in an aviation program or not.
I recommend it.
I was an intern at Northwest and I am still amazed when I think back to how much I was allowed to do as an intern in the ASAP program.
Riding the jumpseat several times a week for months on end is also an excellent way to get a little bit of a preview of multi-pilot operations before you enter that environment yourself.
Delta, United (CAL), UPS, FedEx, and Southwest all have intern positions available.
I'd try and do an internship at that level of the industry. Reason being, when hiring is going on, you might not get your first choice of regional F/O job, but later on, when you meet the minimums to a job you'd like to spend a career at, there's a better chance you'll know someone who can help you get an interview (if you didn't meet someone that could sponsor you at that company yet).