If you are, as the GP said, also distributing it as a web version so people can play it in their normal browser, you already have that problem.

Once you are charging money for the game on a distribution network, you probably aren't providing the full game on a standalone site anymore. Maybe you have a small demo.

Hell, doesn't Steam ship with Chromium (or at least Webkit)? I wonder if it's possible to piggyback on that somehow?

If you did that, any Steam update could break your game.

That's already a possibility if you're using Steamworks (which most Steam games - or at least nearly all I've played - are). Granted, the breakages from a Steamworks-related update are likely far less critical (unless your game critically depends on achievements, cloud saves, instant screenshots, or the Steam Overlay, which would be weird), but it ain't like this would be entirely unprecedented.

Breakage is also already a possibility for games with Linux ports that rely on the Steam-provided runtime environment (I don't know if this applies to Windows or macOS, since I haven't used the former with Steam in years, and haven't used the latter with Steam ever).