Since I misread the question and included RB's that never carried the ball, and now there's a question as to what the answer is, here's what I found. Probably the same as DDD since I used one of the same sources.
1999: Edge + Elias (13 carries, 28 yards)
2000: Edge + Finn (1 carry, 1 yard) + Gordon (4 carries, 13 yards)
2001: Edge + Rhodes + McDougal (17 carries, 48 yards)
2002: Edge + Mungro + Finn (5 carries, 8 yards) + R.Williams (11 carries, 35 yards)
2003: Edge + Rhodes + Mungro + Williams
2004: Edge + Rhodes + Mungro
2005: Edge + Rhodes + Mungro + Carthon (13 carries, 18 yards)
So besides Edge: Rhodes, Mungro, Carthon, Williams, Finn, McDougal, Gordon, & Elias have carried the ball for Indy since Edge joined.