Web client gamepad support

I'm not aware of any roadmap, but you can check currently supported features here.

As for future ones, I guess you can try asking AWS Support

In regards to gamepad support in the AWS WorkSpaces Web client, based on documentation it appears this feature is currently only supported when using the native Windows client and not the web client. However, gamepad support is listed as a new feature that was added in the latest version of the native Windows client. While not available yet, gamepad support for the web client may be part of future roadmaps as the client continues to be developed and configured to allow tighter integration into platforms. For the most up-to-date information on new features and roadmaps, it would be best to check the AWS WorkSpaces documentation or contact AWS Support.

The WorkSpaces Web client streams a Chrome browser session, so any web applications that work in Chrome should also work when accessed via WorkSpaces Web. However, there may be some limitations as Chrome itself does not support Flash, Java or other browser-dependent technologies. For full details on browser and client support for WorkSpaces Web, refer to the online documentation.