My daughter started at a Kid's Day Out program when she was 16 months old. We also have to send them with a sack lunch and they have requested that all foods are finger foods, even for the older kids (3-4 years old).
My daughter is now almost 4 years and eats sandwiches (tuna, turkey, PBJ, etc.) really well, but when she was 2 years old she wouldn't. So, here's what I sent with her (always one main item/meat/pasta along with one fruit and one veggie):
--Gerber turkey or chicken dogs (they come like 7-8 per jar). I would pull out 3-4 the night before and refrigerate them in a little baggie and then pack with a mini cold pack for school.
--Gerber Ravioli (no sauce), is another great finger food and again, I would drain ahead of time and re-package in a little baggie or tupperware. If the caregivers don't mind, you can even send in the package it comes in, if they don't mind opening it for your child.
--Misc. Fruit bits: grapes cut in 4's, peeled apple cut small (little bit of lemon juice and sprinkle of sugar to keep from turning brown), strawberries cut in small pieces.
--Misc. Veggie bits: Gerber green beans, Gerber carrots, lima beans or Edamame. Again, all easy finger foods with little mess.
When my daughter was 1-2 years old, she also had a slight allergy to dairy...so I really had to stay away from it, until now. Otherwise, you can also pack cheese sticks/bites for a healthy part of the lunch. You would need to definitely include an ice pack...but I do that everyday and its so easy to find the small ones at Walmart.
Lastly, there is a new product from Quaker (I believe) that is Granola bites. They come in 3 flavors: Chocolate, Cinnamon & Peanut Butter. They are little Granola "balls" and so perfect for a small snack. Now that my daughter is almost 4, she likes a little something sweet after her meal. So, I usually pack 4-5 granola balls as a treat for her in her lunch. She's very good about eating the main lunch first before ending her meal with the treat.
Hope these ideas help. Let me know if you have further questions. Take care. L.