i also can confirm that they most certainly allow coolers, although we've only brought soft sided ones that would fit under my godson's stroller. The first time we brought him there he was about a year old or so, and was still drinking milk out of a bottle. so we would have some thermos containers with milk in them, and some of his baby food and snacks in the cooler. we kept it with us the whole time under the stoller in case he needed anything. Plus, they HAVE to allow people to be able to bring in some sort of cooler if they need to. there are certain medications that some may be on that need to be kept cool (like certain insulins), and keeping it in a cooler would help them.
i don't think that there is anything wrong or to be ashamed of by bringing in sandwiches. dh and i do this ALL the time to save on money. it depends on our day.

if we eat breakfast at the resort, and plan on staying the park most of the day, we'll bring sandwiches. or we'll hit the park early, bring breakfast bars and fruit

(maybe buy a snack there), and then head back to the resort to make sandwiches in the room. we usually go out for dinner, either a PS at a restaurant or counter service in the parks or at downtown. (one of the luxuries of staying at por is the nice boat ride to downtown - very romantic).
we either make PB&J sandwiches, or i will make them with cold cuts and cheese and then get some mayo or mustard from one of the counter services. we'll buy something from the counter service, like a drink or dessert or something, and eat in a place with a/c.
we don't have this problem, but there are plenty of people with dietary needs, who may be allergic to certain things, or may be diabetic that has to watch their sugar or caloric in take, or with a heart condition and has to watch their sodium intake. they may feel more comfortable eating their own food and know what the intakes are. so, there is nothing embarrassing about bringing in your own food like that.
don't worry about this, i think that it is just a minor issue.

just let her make her sandwiches (maybe that is what she can afford), and then you can get your meals from in the parks. Trust me, SOOOOOOOOOO MANY people do this, i do not think that anyone would be staring at you, or cause you need for embarrassment.

remember everyone is there to have fun.

if her kids see that you're embarrassed, they'll be embarrassed of their mom too, you don't want that.
good luck and have fun!