Hi there Lucie. It sure sounds like you're planning a very magical 21st birthday celebration for your niece!
A great
unofficial website for researching limo transportation can be found at: www.mousesavers.com. Simply type in "limo" in the search box at the top left hand corner of the main page. I utilized Happy Limo on a previous Disney trip and was very impressed with their customer service. And Mouse Savers typically has discount codes for this company listed on their website so, you can hopefully put some extra money back in your pocket for souvenirs.
As for renting a carriage in Magic Kingdom Park, I am not aware of any options other than the traditional Main Street, U.S.A. horse drawn trolleys that are offered at park opening. However, you might be interested in checking out
Disney Signature Dreams. There are some pretty amazing, over-the-top options available with some advanced planning and if your budget allows. Simply contact them directly via e-mail at DisneySignatureDreams@DisneyWorld.com or call 321.939.4662.
Wishing your family a pixie-dusted birthday trip! Please feel free to send more questions my way!