Mail Delivery Service is Act 5 of the third Chapter: Subcon Forest. This act is avalible right after completing Act 1, depending on what contract Hat Kid signs.
After each act Hat Kid will write in her diary about the last mission. The diary can be found in her room under the swimming pool of pillows.
"I don’t get it? He seems super creepy and menacing but he wanted me to deliver mail?
"I have a really bad feeling about this, but he won’t give up his Time Pieces unless I do his weird contracts…"