Try to shadow at a teaching hospital if there is one close to you. I think most of teaching hospitals have a formal shadowing system in place for medical students, and they may allow the rules to be bent for pre-medical students.
I started by e-mailing residency program directors because I figured they would be sympathetic to students, more so than private practice hospital employee physicians. A few never replied. One replied and said he would be happy to help, but then ignored me. I tried calling the guy and his secretary put me in contact with an administrator in the medical education department, who in turn set me up with another, more receptive program director. The formalities -- paperwork, background check, vaccination records, HIPAA training, blood borne pathogen training, fire and safety training, etc. -- took about a month and a half to complete. I'm sure the specific time frames vary from hospital to hospital, but it's probably a good idea to start as soon as possible if you're looking to shadow within a specific time frame.