Sure you can...Water is water regardless of the source. You can use a bucket to fill the bowl if you're so inclined.
Our toilet valve frooze last year so I bypassed it and mounted a ball valve to the wall and ran the line to the toilet inlet. Not fancy but it worked in a pinch.
Why are you working from front to back, instead of back to front? Just curious.
"You wouldn't worry so much about what others think of you if you realized how seldom they do."