Hi Amy,
There is a
maximum stroller size allowed at Disney: 36" x 52". If your stroller is within those parameters, you can bring it into the parks.
As for bringing the stroller on transportation, the answer it "it depends." For any Disney bus or tram, all strollers must be folded, regardless of size. Of course this means that the children can't be in the stroller at the time of transport. Generally children are allowed to stay in fully open strollers when riding the monorail. However, if the transportation system is particularly crowded, a cast member may ask you to fold it. There is also a variety of boat transportation at WDW. On some boats you can roll the stroller on, but the children must be taken out first, on some the children may remain in the stroller.
There is no stroller securing mechanism on any of the forms of Disney transport. You'll be responsible for holding on to it yourself.
I have twins with a slightly older sibling. I made several trips to WDW
when they were about the age of your kids and had a great time. With
some planning, I'm sure you will too.
Good luck!