I need fillers on my face. I would like to know if any otherdoctor can put fillers as long as he or she has a license?

Experience. Would see someone who has experience with this. Anyone may be able to do it, depending on state law, but as it is your face I think you would like to see someone who has done it often.
Anyone can inject. It depends on the state in which you practice, however, fillers typically need to be injected under the supervision of physician, as only physicians can order them. If you feel comfortable with your injector, be it a nurse practitioner, pa, or physician, you should be ok. I would make sure to see some representative before and after pics or a video of the injector working before going to them.