So no nuts / products with nuts
No seeds
No egg / products with egg
No citrus / products with citrus including tomato, peppers etc
No hummus, tahini (chickpea allergy?!!)
No meat from your household
I can see why this is restrictive and why you're finding it difficult. It seems an odd time in the school year to ask this, has it been newly implemented and if so why now?
Personally I would also find this very difficult, especially if school meals do include the allergens. If the rules are that strict they should apply to all.
I would approach the school and say that especially in your situation as non meat eaters you're compromising your children's diet by excluding these food types. It makes it very hard to include protein since egg and chickpeas are off the list, tuna mayonnaise would be an egg derivative ... and who's going to give their kids smoked salmon sandwiches for lunch every day. Ask for verification that they are all banned including the derivatives, if the allergies are that bad could a staff member sit at the table to oversee sharing etc but why doesn't it apply to school meals?
Some other options to consider are more whole food based salads such as rice salads, lentil salads etc... you can get ready to eat lentils / grains easily or make some rice the night before. Same for a cous cous or pasta salad or other vegetables (green beans, broccoli, carrots etc). It doesn't always have to be sandwich based. They can be sweetened if it helps, there are lovely recipes including fruits like apricot, raisins, pomegranate (citrus???) etc. The Japanese do this all the time. They don't really eat sandwiches, you could look up bento boxes online for some more ideas.
If you are doing sandwiches, cheese is fine as the fats are important. You'll just have to use a bread without seeds. You could alternate sliced cheese with e.g. Cream cheese and cucumber. Pitta pockets filled with salad type things can be more fun too. Things like breadsticks make good alternatives. Bananas are also filling and should be ok, as is a low sugar yoghurt and fruits like apples.
If I were you I would meal plan and try to make leftovers from dinner. A real faff that all the extra effort is probably unnecessary (bar the nuts) though.