Supply has black, white and ground wires. Light fixture has black and white wires and a green screw for ground. Green screw connects to fixture itself. I cannot get the supply ground around the fixture's green ground screw and the ground is short to begin with. Can I get another ground wire, connect it to fixture's green ground screw and then wire nut the two grounds together?
Yes. What you propose is safe, code compliant, and good workmanship.
Electrical codes in most locations require that a ground fault wire be either bare (uninsulated), insulated in green, or insulated in green and yellow stripes.