Natalie,
Congratulations on your decision to pursue design.
Most offices will only offer internships to candidates currently enrolled in a design program or even graduates of design programs. Taking on someone with no formal training would be intimidating due to the assumed time the staff would ned to spend to get the person up to speed. I suggest you ping a few people in the industry and start by offering to buy them coffee. If a positive connection happens at one of those coffee sessions, then you could ask to job shadow for a day.
Check the core77 design directory for a list of firms near you:
http://www.designdirectory.com