Yes,
every action available in the User Interface is possible via the REST API. Sometimes the easiest way to find the correct REST endpoints to use is to open your browser's DevTools and watch the Network tab, as it'll include every request made to the REST API each time you perform an action in the UI.
In terms of approving/rejecting items, you'd have to use several endpoints.
First, you *MUST* have a Workflow Item created..
It's possible this is already happening automatically though, if you have a Collection with workflows enabled.
There's a lot more that can be done here, obviously, but that's the basics of the "flow". I'd highly recommend using your Browser's DevTools to look at each request made when performing these actions from the UI...it's possible I forgot a step or simplified something here.
If you have further questions, feel free to ask on this list.
Good luck,
Tim