You cannot bring food, drinks and large bags or luggage inside the museum. You will have to leave them in the appropriate lockers at the entrance.
Food and drinks are forbidden inside the museum for conservation purposes. The visit only lasts 15 minutes, so they shouldn't be needed either way.
No, there is no strict dress code for visiting the Last Supper, but it is recommended to dress modestly, considering that the artwork is in a religious building.
To enter the Last Supper Museum you need a valid ticket and your ID. If you are eligible for a reduced or free ticket, you need all the connected documentation as well.