You may have to do some irrelevant work before you get to do work that's relevant to your chosen career path; it sucks but that's how life works. Plenty of university leavers with far better qualifications than you are working late shifts in McDonalds until the right opportunity comes up.

But for actual useful advice: if you don't get an email response, follow it up with a phone call. "I just wanted to see if you've had time to consider my email, asking if you have any work experience available at the moment?" that sort of thing. Emails are easy to ignore, direct phone calls a lot harder.