I'm sure that it will not be a problem. Many people take pack lunches and their own drinks into the parks. Although be prepared to have your bags searched on the way in. I have seen people have alcohol confiscate but not soft drink or food.
We took loads of different food into the parks without a problem. I remember seeing a post somewhere here about Disney cooking special meals for people with allergies. I think it was on one of the American threads.
You will have no problem, our son has an eating disorder, so we have to not only take all his food into the parks with us, we have to pack it all in our suitcases and take to florida with us LOL!!!!