Yes, you can omit "away" and the meaning would be the same.
However, adding "away" in this context draws attention to the fact that the action is ongoing and uninterrupted, it has been happening for quite some time and/or will continue happening for a while longer. Some other examples:
"He was sat in the corner, thinking away"
(He was lost deep in thought, or had been thinking for some time)
"...to the soothing sound of a fireplace crackling away in the background"
(The crackling of the fireplace is a constant, ongoing sound)
"When she's in the zone, she sits for hours, painting away"
(Losing track of time, doing nothing else, uninterrupted)
The 'away' draws emphasis to the fact that the action is free to happen, unhindered. It can also be used to give permission or encourage an action. For example, a teacher might say to her art students "Paint away!" to tell them that they're allowed to begin painting.
I hope that helps!