well no you can not write it with apostrophe as it will change the meaning. You are right adding "s" will change your name itself. I can suggest some options like: The Palermo family from Palermo family or you can actually add "s" as people who know you obviously know that your name is Paplermo nd palermos and hopefummy they will understand the added "s".
An apostrophe is never used to form the plural of a family name.
You shouldn't worry; no one's going to think that the card has been signed by an individual or organisation called 'The Palermos'. If it still bothers you, you can always sign the card 'The Palermo Family' or 'Jack and Jill Palermo'.