You can do whatever you program your bot to do. If you want it to interact with your own API then it’s entirely up to you to write the code that will allow it to make those requests your API requires.
If you’re using a bot developed by someone else, then you’d have to look at their documentation to see what features their bot has and if it has the capability to send requests to other services external to Twitch.
I wrote a IRC bot in less than 5 minutes and managed to use it to call underlying frameworks in very few lines of code, just experiment with it as said. It’s pretty easy to figure out once you know each programming language that you need to depend on for it to work.