If it's HTTP(S) you should be able to use any kind of packet dumping tool available on Google Play. It will run as a local VPN and you should be able to dump requests that come from the app. If the protocol is lower level you'd probably want to use Wireshark to dump traffic data. You can connect your phone to an access point hosted on your computer that Wireshark can intercept data from, or it might be easier to use some Android emulator on the computer.

I've done reverse engineering work in the past on games with online functionality (e.g. 1, 2), but they've usually been older and smaller ones where the traffic is unencrypted and could easily be dumped and inspected. Or reimplemented by looking at the client. So I can't explain all the steps in detail for doing it to a modern game, but when you're dumping encrypted traffic you will be basically doing a man-in-the-middle attack on yourself. Which is not too bad as you of course can trust yourself and can install the certificate that you will be using to decrypt and reencrypt traffic, but if the app in question does additional checks for the integrity of the server (e.g. pinning a specific certificate it knows the real server will use) that would probably cause some issues that will interfere with your ability to use the app to dump the necessary data. You would need to crack open the game client to be more lenient in that case.