Many controllers are supported, a good list of “sure” working ones for desktop platforms here is here: https://github.com/gabomdq/SDL_GameControllerDB/blob/master/gamecontrollerdb.txt - although on Linux and Windows, any Xbox compatible USB controller will surely work, as well as many others.
However, this does not suppress arrows.
BTW, if you’re on Android, controller support is somewhat limited; it seems like it is also necessary for Android to recognize its layout, which it does not do for a controller I have (a cheap Xbox controller knockoff from Walmart). If you want to suppress the touch controls on Android, there’s a way but it involves ADB (there’s currently no setting for this) to change the settings JSON file to set “touch_force” to “false”. On the web, similarly your browser needs to be able to map it to the W3C standard layout, which is basically Xbox style.
The game requires at least the features a typical NES controller had; however it is implemented such that you can use any stick or d-pad on your controller, they’re all activated.
BTW you can port your savegame from Android or web to the desktop version as well, if desired; see instructions on https://divverent.github.io/aaaaxy/userguide.html under “data storage”.