I volunteer at one, YMMV depending on state laws.
Most of the free clinics here are in some way sponsored by local hospitals or churches. Hospitals will donate labs or services such as surgery that the clinics can refer them to. Medications can be purchased through the hospitals buying power at cost. Churches will donate floor space, money and/or volunteers (usually non-clinical). There are also philanthropic organizations that will donate money. All of this donation usually comes with strings attached so you need someone in charge who is a good negotiator. You also need someone who can try to get med companies to donate product, and/or assist patient in filling out medication charity work.
You have to register with the state as a charitable pharmacy, and there must be a pharmacist "in-charge". You will be inspected by the BOP every ywar and everyone must have a tech license to volunteer in the pharmacy. Find a group of docs and nurses that want to volunteer and that are reliable, some will come from the hospitial that assists you. Many of the docs will bring in med samples from their clinics for you to use. Ideally you want to get volunteers before they get into professional school that will want to stay and volunteer after graduation. Have people around that speak other languages. We have a lot of only spanish speaking patients, and we have to do a lot of translation through family and pantomiming.
All the patients here sign wavers each time saying that they hold the medical providers blameless, which of course doesnt protect 100% from ligitation, but I believe there are laws in place to protect medical volunteers.