Played a game of Legion a week or two ago and this issue came up. If Vader moves twice then uses force push to pull a character into base contact can he then apply relentless to melee attack that unit?

One guy argued that the force push didn’t count because it was a free action so the attack was “after a move.” The other guy (me) argued that if you move, then take an action, it’s not then “after a move” it’s “after a free action” and therefore you don’t get to apply relentless. Any thoughts?

You had it right. "After" always means "immediately after". If you do another action in between (even a free action), then the trigger window has passed. From page 11:

Quote

If the timing of an ability uses the word “after,” that ability’s
effect occurs immediately after the described timing event
has occurred.