If you're not in a hurry, you could take Amtrak from TPA to KIS, using Uber to get from the airport to the train station in Tampa and from the Kissimmee station to WDW. There's only one train a day, which is scheduled to depart at 5:20 pm, but it often runs late. But the fare, if purchased in advance with the no refund option, is only $11 a person.
But you'll need to be patient and flexible, as the train is late more often than it's on time. Usually it's within an hour or so, but within the past 30 days it was 3 hours late to Kissimmee once, and a whopping 6 hours late another time. The former was probably a mechanical issue, the latter likely a run-in with an automobile at a grade crossing; when that happens, everything stops until the police complete their investigation.
Seats are guaranteed but not pre-assigned, and the train originates in Miami, so there's also a very good chance that you might have to split up on the train. The train attendant will try to help find seats to keep children with a parent, but there's no guarantee.
Going the other way is more risky. I wouldn't try it unless you have an evening flight and you're willing to spend a good part of the day exploring Tampa. The return train is scheduled to arrive at TPA at 12:55 pm. Over the last 30 days, the train arrived in Tampa 5 1/2 hours late twice, and 90 minutes to 2 1/2 hours late 7 times. Better have a Plan B, and activate it if the train is running more than an hour late into Orlando.
https://www.amtrak.com/home.html