Yes, FIGHTER is a Scrabble word!

Letter values in Scrabble US: F 4, I 1, G 2, H 4, T 1, E 1, R 1.

Is it a valid word?

FIGHTER is a valid word in the official Scrabble dictionary (US), the UK Scrabble Dictionary and Words with Friends (ENABLE).

someone who fights (or is fighting)
a high-speed military or naval airplane designed to destroy enemy aircraft in the air
someone who fights for a cause

Also: fighter aircraft, attack aircraft.