Yes you can! We brought our soft sided cooler that is on wheels into TL on two different days in Nov. and we ate ham sandwiches, chips and soda. Some family members who just happened to be at WDW at the same time meet up with us that day and they didn't bring a cooler so when it was time for lunch they went to get in line for food while we went and fixed our own sandwiches and we were just finishing up eating when they showed up with there food tray! So this was also a big time saver.
BTW we didn't share our food as it wasn't a planned thing and we only had enough ham and cheese for the 4 of us. We did share chips and juice boxes though! I just thought that might sound like we were being mean and not sharing with our relatives.