Hi Jennifer,
I am glad you came to the Disney Parks Moms Panel with your great question! Spilled drinks are never fun, so as the mother of three, I completely understand your concern. Great to hear you are thinking ahead for everything, including these small details.
I have great news!
All dining locations, except for those at Disney's Animal Kingdom, offer lids and straws for drinks. Many even offer bendy straws, which are my favorite for the really little kids, since they tend to tip them if the straw is too long. For the safety of the animals in Disney's Animal Kingdom or Disney's Animal Kingdom Lodge and Villas, plastic straws and lids are prohibited anywhere in the park. They do offer paper straws, but absolutely no lids. So, there is still a chance for spillage there. Keep this in mind if you are concerned. This may mean packing at least one sippy cup for that purpose. However, it is only a small concern since the other parks offer lids and straws. Also, if you plan to bring in juice boxes or pouches that have plastic straws, the straws will be confiscated when they do a bag check at Disney's Animal Kingdom. I know, it happened to me. I had not even thought about that when I packed them for the day. Again, this is for the well-being of the animals.
I hope this helps you with your packing and planning. Have a magical trip and please feel free to come back with more questions if they come up prior to you leaving for you vacation.
~Allison L