This is going to sound glib, but you can just keep using what you have now.
If you take a look at [1], the maintainers of request aren't "deprecating" it in the sense that they won't touch it ever again. It really just means that there won't be any more active development on it, no new features or breaking changes will be accepted, and some housekeeping had to be done to allow it to accept security fixes without relying on one specific person to be around.
The code will still work (and probably for quite a while), and security issues will still be addressed, it just won't be changing much if at all from now on.