Hi, Jennifer! You absolutely can bring a soft, bag style cooler into the theme parks with you! It's a great way to keep snacks handy and save a little bit of money that would better be spent on souvenirs!
As long as the cooler can be carried, you should be fine. You aren't permitted coolers that have to be wheeled. Also keep in mind that you cannot have glass bottles or things breakable of that nature in the cooler. You will have to go through the security bag check with the cooler, so keep it handy for a quick inspection as you enter the theme parks.
Many people ice up their coolers at the hotel before they leave, but if you are planning a full theme park day, keep an eye on the melting ice. If you plan to keep perishable items in the cooler (or items that could melt - chocolate covered granola bars, chocolate candy bars, some fruit snacks), they still risk damage from the heat as the day progresses.
You can get specific details on the cooler size limitations by reading Disney Mom Brigitta's fabulous answer, which is located
HERE.
I hope you have a great visit to Walt Disney World!