Hi Erica!
Welcome to the Disney Parks Moms Panel! It's my pleasure to help a fellow Texan.
Magic Kingdom Park does have three sizes of
rental lockers, with the biggest size being 17" x 22" x 26". While there is no place to leave or check larger items inside Disney's Theme Parks, what I'd do is request to be dropped off at
Disney's Contemporary Resort. You'll be able to store the carseat with the luggage service at this resort hotel despite not staying there as the items stored are tagged in an independent system. And bonus, the Contemporary is closer to Magic Kingdom Park than where taxis and app-based ride hailing services are allowed to pick up and drop off their Theme Park passengers. They can only visit the Ticket and Transportation Center, which would require you to take a
Monorail or Ferryboat to Magic Kingdom Park. From Disney's Contemporary Resort you can either take the Resort Loop Monorail to the park (the station is inside the hotel on the fourth floor; Magic Kingdom Park will be the 4th stop), or there is paved walking path that takes about 10 minutes max. My favorite part of The Contemporary plan is both the monorail station and walking path each have their own dedicated security checkpoint so you'll bypass the giant one at the Ticket and Transportation Center.
Another great option if you're staying at a Disney Resort Hotel is to use
Disney's Minnie Van Service. These vehicles come with the complimentary use of two carseats AND they are allowed to drop guests off right in front of Magic Kingdom Park.
We'll See You Real Soon, Erica!
~Kristen