Due to Parks Canada Eligable Resident regulation residential property is to be resserved for workers and students who need to live in the park full time. The Town of Banff Residential Leases webpage provides more details.
Here is the key paragraph/line for the Town of Banff website:
Residential leases in Banff restrict occupancy to eligible residents as defined in National Park regulations. The eligible residency provisions ensure that community residential lands are available exclusively for community use, rather than recreational or second home purposes.
If you rent a vacation rental inside of Banff National Park it is possible that it could get shut down prior to, or during, your stay. You can rent hotel rooms inside the park and it could be possible that a house has a zoning other than residential, such as, commerical use which would make it possible but make sure you ask a lot of extra questions before you rent a vacation rental home inside the park.