The expression “på vei” conveys that she is in the process of moving to some particular place, but it doesn’t say anything about how – she might be walking or canoeing, for example. (The phrase might also mean that she is pregnant.)
The expression “på veien” would instead simply mean that she is physically located on a road.
I put, "Is she on the road?", and it was marked as correct even with no 'en' on the end of 'vei', so it must've been taken to have the same meaning. I guess 'on the road' could mean not only physically located on it, but 'on the way', or one could ask if a young person was 'on the road' in terms of allowed to drive yet. Not sure what you'd use for that.