I don't know the exact time, but it's no later than 8:30 a.m. The thing is that buses don't run to all resorts until later, but they will be running to some nearby resorts.
If you're just trying to get to a park, it really doesn't matter which resort you go to; it just might take longer if the resort is out of the way. POFQ is suggested because buses run there pretty regularly in the a.m., it's fairly close to DTD, it's also close to Epcot, DHS, and MK, and perhaps because of the aforementioned SSR issue.
If you're trying to get to a park for an ADR, it's probably best to use the hotel shuttle. Just find out what the schedule is ahead of time, and allow time for transfer by monorail if it's the MK.
If it's an early breakfast, before the hotel shuttles are running, a taxi is best.
If you're going to a resort later in the day, and you want to walk to DTD for a Disney bus, remember that Disney buses out of the Marketplace will stop at PI and TL on the way to the resort, so it can take an hour or longer.