How can I add a program to the "Open With" menu on a Mac?
Do you mean the submenu you get when right-clicking on a document? If so, it's generated automatically by Launch Services based on the document types your applications claim to be able to handle. If you look inside an application package (right-click the application, and select Show Package Contents), inside the Contents folder there'll be an Info.plist file with various information about the application, including an array of document types it can open (see Apple's dev documentation here).
TLDR; if the application handles that type of document, it should already be listed; if not, I don't know a way to add it manually.