Jarvis is not a Avenger. He might be regarded as being one of their own but he is in really their butler/housekeeper/caretaker and though he regard the Avengers as close confidantes, he himself would never think of himself as an Avenger member.
I think, Jarvis was made an Avenger in the first issue of the Mighty Avengers series where Hank Pym was the Wasp!(Mighty Avengers - Dark Reign No. 24)He fought alongside Hercules, Amadeus Cho, Jocasta, Hank the Wasp Pym, Stature, a version of Vision, Hulk, Scarlet Witch and USAgent