I know you should always wear a suit when applying for jobs and internships. I want to get into this program at the university I am currently enrolled at, but I need to get some volunteer experience prior to applying to the program.

Really depends on the position you are applying for.
Just because it's volunteering doesn't mean your personal standards should slip/

What do the volunteers do, and what do they wear when they do it?

The rule is to always dress up, not down. Find out what the volunteers usually wear on the job and go up a notch from that. Or, when in doubt, wear a suit or sport jacket or nice pullover sweater with a dress shirt and tie underneath (or some such outfit).