Kimberly,
Disney is very good at offering event hosting of this kind. They have a very wide range of facilities, from the gigantic and expensive (yes, you can rent out an entire park!), to smaller and more affordable.
Also, yes, you can rent select characters to come to visit your party. Again, this can be very expensive, but with a wide range of prices, you don't want to assume in advance that it is our of your range.
The first people most people contact are the
Disney Event Group or the
Disney Meetings group. But these are really more for corporate events.
Probably a better resource would be to e-mail or call the
Disney Group Services at disneygroupgetaways@disneyworld.com or call 407-828-3318. They should be able to assist. You can also just fill out the "
Contact Us" form at the Walt Disney World site with your question. They can forward you on to that group that best suits your party size.
Probably one of the easiest things to do would be to get a convention room at one of their resorts that have a convention facility. But probably more to your scale, and possibly budget, would be to work with a Disney restaurant that has a private room. While most are available at signature restaurants (Jiko, Hollywood Brown Derby, and the California Grill have such rooms, to name only a few), there are rooms at some of the more affordable venues. One often overlooked on is at the Wave restaurant in the Contemporary Resort, but there are many more.
Best of luck!
Have a magical vacation!