The Israeli military said no one was injured in the Jan. 20 attack but a vehicle was damaged, the Jewish Telegraphic Agency reported.
The military said the goalie, Omar Abu Rois, 23, is affiliated with the militant group Hamas and works for the Red Crescent.
Munzar Abbas, 41, a Red Crescent security official and an officer of the Palestinian intelligence service, allegedly supplied AK-47 rifles used in the attack, the military said.
All those arrested are residents of the Amari refugee camp near Ramallah, where the shooting occurred, the military said.
The military said the suspects had planned similar attacks in at least six other locations.