Surprised to see people taking taxis and Uber. I always use WDW transportation. Now, for your case. My experience has taught me the bus to Hollywood Studios usually goes to the Polynesian first, then the Grand Floridian, so it may be a little crowded but that early you should be fine. I would assume you can also try taking a bus to EPCOT or if you want to get adventurous, I believe the boats go from Magic Kingdom, Grand Floridian, Polynesian, so you can ride a boat to the Polynesian and walk to the Transportation and Ticket Center to get the monorail to EPCOT.
You shouldn't have an issue using WDW transportation, maybe ask someone at the main desk of the Grand Floridian what the transportation schedules/timing is usually like that early, but you should be fine.
I don't know, I find Uber pretty weird, I mean, you are getting into some random person's car and trusting their ability to bring you somewhere. I know a taxi is similar, but at least they have some sort of regulations and it seems like it is more structured.