Here's how late check-out works:
It's controlled by Housekeeping, NOT by the front desk. If you ask at the front desk, they'll ask you to wait while they call Housekeeping and ask.
You can't really request it until the morning of check-out day. You could try the night before, but don't be surprised if they say to call back in the morning.
IF you are granted late check-out, here's the fee schedule:
Up until 1 p.m. -- no charge
1:01 - 3:00 p.m. -- 50% of nightly rack rate
After 3 p.m. -- 100% of nightly rack rate
DVC resorts do not allow late check-outs, except in exceedingly rare situations. They tend to have a higher occupancy rate, and they take longer to clean than regular rooms (because people staying on DVC points do not get daily housekeeping service, the room will often be messier).