Hospitals are very used to shadowing and volunteer requests, usually from people wanting to get into medical school. They have a process in place.

Here's the process: Make an appointment with the lab director, and get their approval first. Then, talk to the hospital education or whichever department they refer you to. As long as you have the approval of the department, it shouldn't be an issue. They also usually have a medical volunteer coordinator that can help.

Shadowing, you will only be able to observe. Don't get in the way. Take notes and write down your questions for later. Be respectful and punctual.

Volunteer, you're going to be doing very limited work. Delivering specimens, centrifuging, etc. Pay attention and make good use of downtime to learn more about the job.

Consider either situation to be a two-way job interview. They're considering if you would be a good coworker, and you're considering if this is a job field you would really want to be in.

Source: I've hired former students and volunteers, and assisted people in getting their feet in the door before.