Could a graphene battery / supercapacitor be used to power a railgun?
The problem with capacitors is not quick discharge capability, it is energy density, and particularly the energy density of types of capacitor that can discharge very quickly.

There are apparently some batteries on the market that use graphene in their design. However, the use of graphene is a research area that some are claiming has great potential, not something that has a huge proven potential.

I believe that there are superconductor-based inductive energy storage systems that have been installed, proven and currently offered for sale. However they are of the scale that would be transported on a large truck or rail car, not something that has potential for anything near hand-held.

It is very doubtful that a useful hand-held rail-gun is any closer to reality than a nuclear fusion powered automobile.

Added information about inductive energy storage

The difficulty with small scale superconductor-based inductive energy storage systems is that there is a substantial balance of system (BOS). The BOS is all of the required components and equipment in addition to the basic energy storage component. Refrigeration equipment and insulation is required to get the superconductive material cold and keep it cold. There are also charging and discharging or energy transfer systems.