Last I heard,
a) You can only use Minnie Vans for airport transportation if you are staying Club Level at a Disney hotel.
You can use Minnie Vans between Disney hotels and Disney parks.
In addition to Disney-owned hotels, this includes Swan, Dolphin, Swan Reserve, Shades of Green, and Four Seasons Resort Orlando at Walt Disney World Resort.
Minnie Vans have 2 car seats. The driver installs them for you.
Minnie Van info here:
https://disneyworld.disney.go.com/guest-services/minnie-van-service/
The website says 6 guests, but I don't know if that capacity is reduced if 2 of the guests are in car seats.
b) Lyft does not have vehicles with car seats.
c) Uber technically has vehicles with car seats, but it's only one car seat, and there aren't very many of them. I personally wouldn't count on getting two of them when you want them.
For airport transportation, you can look into private towncar services. There are many threads on those options. You would make a reservation, so you would know that the vehicle would be there when you need it.
You also can look into
https://www.mearsconnect.com/ which has airport shuttles.
You could use regular Disney transportation (bus, boat, monorail) between Disney hotels and Disney parks.
Depending on the ages of the kids, some people use inflatable booster seats, such as this one:
https://bubblebum.co/?srsltid=AfmBOoqCLJHlwH-opsJLWmphbL_R_mkSz7YY5iCLi2XGGqEHae-pRUa0
On a typical park day, I wonder if all 4 adults and both kids will be on the same schedule all day long. I would think that even if you all start out together in the morning, some people might want to go back to the hotel before others. So you would not necessarily be looking for transportation for your entire group all the time.