Scooter rental at Disney World is $50 day + a $20 deposit. Deposit is refunded if you turn scooter back in at the rental counter (if you just abandon it, you lose the $20 deposit...so it would cost your $70 total)
At Magic Kingdom, the scooter rental counter is over to the right, before your enter the walkway into Magic Kingdom...so after coming through the ticket turnstyles, go over to the right. You should see some
ECVs parked in front of the counter there.
Do keep in mind if you're getting there just before the party starts, it can take 15 minutes to get a scooter (filling out paperwork, paying, waiting in line, etc.) And unfortunately, there is always a chance that no scooters will be available late in the afternoon. So you may have to rent a wheelchair instead.