The college I am enrolled at offers internships at local hospital before I graduate. I have to complete a specific amount of volunteers hours before I can get into the sonographer program at my college. I was looking online at job positions in ultrasound and most of them require 1 to 3 years of clinical experience. I have at least 6 local hospitals I can volunteer in radiology departments. I looked for places to shadow a ultrasound technician, but I can't find any offered in my state. I'm trying to intern, but the internships also require some clinical experience.

Internships in hospitals usually aren't published these days. You have to go schmooze people in person at local hospitals, get to know them, be friendly and eager, ask to talk over coffee or lunch in the cafeteria...
They will be busy, so suggest a specific day and time to talk, maybe near lunch or the end of a shift.

I think that one of your questions is- will the volunteer experience be counted toward meeting the minimum experience requirements for an ultimate full time (or part time too I guess) position?
My opinion is based on how that was handled in the state agency I worked for - for 35 years. The advertisements for positions listed the specific components of experience that the particular position required and the number of years of that type experience required. There was a general statement included on the ad stating that volunteer experience (of the type specified) would be counted year for year toward the required experience.
However, every company is different and the only way you can get an idea of the likelihood of it being counted is to check with one or more of the companies that you would possibly seek employment with, or try and find the official requirements for experience in different positions on their website possibly. And people and rules could still change before you actually applied in the future.
There is no good reason (IMHO) why volunteer work of the right type should not be counted toward required experience for any position. But, you mentioned following a tech around or a similar description, that would be a different proposition and not hands on experience as is likely required. I could sit in the Dr's waiting room watching the office personnel work behind the counter all day but it wouldn't make me an experienced medical office worker. (Actually, they would throw me out in the cold anyway pretty quickly.)

In general, it may count, it will depend on the agency that is hiring.

You need to call the hospitals that are in your area. If there are 6 hospitals where you live, surely at least one of them has an ultrasound tech that you can shadow or can direct you to where there is one. You can't just rely on info and postings on websites. You have to call and ask around. Talk to your teachers too and ask if they know someone.
Also do a Google search of healthcare staffing agencies in your area and give them a call. They usually have a lot of connections and might be able to find you an internship.

I don't know your location, but speaking from my location, many hospitals no longer allow volunteers of this level. If it is a reputable college they will provide clinical experience and assist with placing a person in an internship.