Logging time in chat isn’t too hard, there are two main ways to do it - by having an IRC client that handles joins/leaves, or by polling the API (https://tmi.twitch.tv/group/user/ryevenge/chatters the TMI API is undocumented and may change without notice)
Making your web application from there really isn’t that different from any other application. You will need to find out how to make an application where user login/sign-up is done with OAuth2. Perhaps you can find a framework which does most of this for you out of the box (or seek out some tutorials).