Mayo and any refrigerated lunch meat will spoil very quickly, if not cooled properly, and coolers mostly cannot hold appropriate temperatures for very long. If you cut canned or dried meat for sandwiches (e.g. canned ham), it will spoil as well. Cheese might be suitable, depending on its moisture content, i.e. hard cheese lasts much longer than soft or cream cheese.
Basically, anything that is stored at room temperature in your supermarket is suitable. However, beware that many canned or jarred items must be refrigerated after opening, even mayo.
If you use your cooler and plan your first lunch after five or six hours, you can make cheese or veggie sandwiches in advance with butter or a spread that is usually stored at room temperature. Salads or cooked veggies (e.g. with rice) are also good, if you keep the sauce / dressing separate. I personally don't like ready-made dressings, but do not use perishable food such as yoghurt, when you make your own dressing. You can experiment with cooked meat or other items / meals, but keeping these in a hot car in summer is risky, even with a cooler.
For the second lunch:
- canned / jarred products like fish, meat, vegetables, dips or sauces, with a couple of bagels or other bread and a fork and spoon
- whole fresh veggies (not cut) combined with "summer sausages" and or a canned / jarred dip
- whole fresh fruit (not cut) combined with nuts, seeds, dried fruit and / or granola
- some cereals and oatmeal can be made without cooking, and you can prepare it with juice instead of milk or use UHT milk. Use a plastic container for a portion of cereal, maybe with nuts and dried fruit and pour some juice on it whenever you're ready - no long prep needed.