Hello, Brona!
Let's get the bad news out of the way straight off. Unfortunately, buggy boards are not permitted at
Walt Disney World. If you take a peek at the official
Property Rules, you'll see prohibited items include any "trailer-like object that is pushed, pulled, or towed . . . by a stroller." This includes buggy boards. Also, in many cases, adding the buggy board to the back may bring the stroller out of the maximum measurements a stroller can be in the theme park, which is 79 cm wide and 132 cm in length.
But let's look for a silver lining here, shall we? I travel to Walt Disney World all the time with my three grandbabies, who are three, two, and not quite one. Disney World just rolled out some brand new
strollers that are available as rentals during your stay. Depending on the ages of the children, you may be able to rent a double stroller and either rotate someone in and out (taking turns) or perhaps use a baby carrier part of the time.
The other option to consider if you'll have another adult with you might be to bring your stroller without the buggy board, then rent a single stroller at the park. It may not be an ideal solution, but it's a solution.
My daughter uses a Zoe brand stroller - The Trio+. I'm not gonna lie - it's pricey at over $500. But it fits the size requirements, and we've gotten so much use out of it, both at Disney and at home.
I hope you find an option that will work for your family, Brona. Good luck!
Anna