The reason for using "could" in this sentence is that it is a polite and appropriate modal verb for making a request. Modal verbs like "could" are often used to ask for assistance or make polite requests because they soften the tone of the request. Here's the explanation for each option:

1.) "Could" is a polite way to make a request, and it is commonly used in such contexts.

2.) "Should" is used to indicate a suggestion or recommendation, not a request. While it can be used for requests, "could" is more polite.

3.) "May" is used to seek permission, not to make a request. It asks whether you have permission to post the letter.

4.) "Shall" is more formal and is not commonly used in this context. It can sound overly authoritative and is not as polite as "could."