You will get a lot of debate on the Families board, but we have been taking the kids on Disney transportation without car seats since they were babies. And an occasional taxi or uber as well. We live in NYC, where we use public transportation ALL the time - subways, city buses, and to a lesser extent, taxis and car services. Then again, I am also the person who has taken my kids on 50+ flights without car seats.
If you're staying on site, Disney transportation is fine. We have taken DME with the kids many times. This last time since we had a split stay (Dolphin/BLT) we used
FL tours which provided a booster and car seat for the airport transportation. We took uber one night to go to AK, a bus one day to go to AK, and then walked to the other parks (EP, HS, MK).
It is up to you, of course, but millions of people have taken their babies and toddlers on mass transit - at WDW and elsewhere - without car seats and have been fine. Nowhere is it required (and at Disney, not allowed on DME) to have your kids in a booster or car seat on mass transit or public transit. As you see from above - to/from the airport in a car, yes, we get the car seat and booster, but for a short trip at relatively low speeds in clear weather, we made the assessment that the very small risk was one we were willing to take.