Button cells for arduino uno
Determine the current necessary to run the servos from their datasheets. Then make sure they can handle the higher voltage. Then make sure the 9v cell has the capacity to run it as long as you want it to (broad estimates will do). A 9V will likely have between 400mah and 600mah. That's milli-amps times hours if you didn't know. So for instance, if the servo draws 0.5 amps at 9V then it could run (be actively on) for about 1 full hour before crapping out.

As long as you know how to drive the Uno separately from the load then yes, this idea should work fine. You'll need either relays, bjt's or mosfets to take the signal from the Uno and drive the servo in this manner. I'd recommend N-MOS's to do that job.

Lastly, more details in your question would result in more details in answers.