there are three parts to your problem 1. recieving mail and processing it. Since you are planning to host it by yourself answer would depend on what are you using for hosting. Solution can be as simple as reading a file(i.e. recieved mail) and extracting relevant fields using whatever language you know python/shell ecript etc. This part would be highly specific to what you use as your mail server.

2. server that responds to the api request made by chrome extension. This is where you can use django it would provide basic security as well as cater to the api request made by your extension here any server would do as your use case is quite simplistic. It should be few lines of code depending on your familiarity of language.

3. Chrome extension this would be in javascript.

are you developer? or do you have access to developers? if yes than it should be straightforward.