Can I use an amateur radio for a science fair? If so, do I need to prove that I have an active license?
The short answer is probably. Since amateur radio is a scientific hobby, that would be a great science fair project.
You should check with the science fair administrators before you do a lot of preparation, but I cannot fathom why any science fair would reject an amateur radio exhibit. I suggest you make certain that you present your exhibit as scientific somehow, and not just present it as (for example) "Hey, I can talk to the local net, and that's fun", or present it only from the aspect that hams sometimes use it to provide a public service (like storm spotters, or disaster communications when no telephones are working).
In any case, bring a copy of your license. You can print it from the ULS page, I believe. My wife and I keep a copy of our FCC licenses with us at all times.
It will depend on the science fair, as well as who the fair is affiliated with (if anyone). For instance, the Intel International Science and Engineering Fair says:
Projects must adhere to local, state and U.S. Federal laws, regulations and permitting conditions. In addition, projects conducted outside the U.S. must also adhere to the laws of the country and jurisdiction in which the project was performed.
Adhering to laws means you'll need to have a ham license to transmit (or find a ham who will is willing to act as control operator at all times when you transmit).
All projects must adhere to the requirements of the affiliated fair(s) in which it competes to qualify for participation in the Intel ISEF. Affiliated fairs may have additional restrictions or requirements. Knowledge of these requirements is the responsibility of the student and Adult Sponsor.
So even if you have a license or a control operator, you'll need to check with the fair itself.
Finally, be aware that there may be additional requirements when using RF:
A risk assessment (documented on Form 3) must be conducted when a student’s project involves radiation beyond that normally encountered in everyday life. Non- ionizing radiation includes the spectrum of ultraviolet (UV), visible light, infrared (IR), microwave (NW), radiofrequency (RF) and extremely low frequency (ELF).