Rosie8801:
We have a 23-month old daughter that is allergic to Milk products (cows milk, milk proteins, etc) so we know all about it.
We just came back from 9-days in WDW and I'll tell you what we did ahead of time and what our experience was.
Before we left we sent an email to the Special Diets email address for the Magic Kingdom. The email got to an Executive Chef for the Magic Kingdom and they sent us a bunch of information for their parks detailing what does and what does not have dairy/milk, etc as an ingredient. It was really nice to get that as we had a lot of questions about mainly the counter service meals and the attachments they sent in email really gave us some good info. They also sent us an word document showing our current advanced reservations with the ALLERGY notation clearly identified on each one so that was nice as well. What we found out from that info was there were a couple of snacks within Disney that she could eat (i.e. Non Dairy Chocolate Chip Cookies, some frozen fruit snacks, some bread/rolls, etc) that we would never have found without that info but the counter service meals would probably be somewhat tough unless she ate french fries every time (which she did eat on occasion.)

It was nice to have the info though and it helped us tremendously.
When we got to Disney we checked into the sit down meals and each cast member always confirmed that we had a Dairy Allergy person with us and that they would have the Chef speak with us at our table. Each time the Chef came over (and every one of them was super nice) and detailed what our daughter could eat. Here's some examples of restaurants and what they did:
1. Boma (AK Lodge): The chef walked my wife up to the buffet tables and pointed out what our daughter could or could NOT eat. They also brought her Toffuti (non Dairy) Ice Cream for dessert with little Mickey Sprinkles on it which she loved!
2. Kona Cafe (Polynesian): The chef came to our table and recommended the Grilled Chicken and Pasta meal. The pasta normally had cheese on it but he had them cook it without the cheese. He also personally brought her a no-milk dinner roll (I think they're called Tapioca Rolls) which she thoroughly enjoyed and also Toffuti Ice Cream for dessert.
3. Hollywood & Vine (Breakfast with Playhouse Disney): The chef took me up to the breakfast buffet and showed me the items she could eat...and made it known which to stay away from. He asked what she likes to eat in the morning and I told him she really likes waffles. He indicated that they did have VANS waffles in the back (which are non-dairy frozen waffles) and that he would toast one up for her. She loved it! The chef again was very nice and willing to do whatever was necessary to make our little one happy.....he even brought her an extra Silk Soy Milk in case we needed it later in the day!
Every other restaurant was the same.....Chef Mickey, 1900 Park Fare, Garden Grill (EPCOT)......each time the chef came out to walk us through things. It was great.
Regarding counter service meals, it was a little tougher as there was simply not that much that we found that was dairy free. The chicken strips all had milk in the breading......the rolls for the burgers had milk.....the desserts for the most part had milk.....etc. So, what we did was we FedEx'd a package to our hotel a few days before arrival with all of the breakfast/lunch type items that we know she can eat and also, because we rented a car down there, we also stopped off at the local grocery store for Soy Milk, etc. That made it much easier on us.
I would recommend you make contact with the Special Diets folks at Disney (which will most likely be one of the Executive Chefs at the parks) and let them provide you some more info like what we received. It will really help you plan out your meals and make it a smoother ride for everyone.
Good luck and feel free to contact me through this site or my blog if you need more info.
Brian