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.
"på veien" indicates that she is physically on the road. "på vei" says that she is travelling/moving along (using any form of transportation you can imagine).