k1ne

You could certainly use it to do that, yes.

Depends on why you're not connected. If you disconnected because Steam went down or you otherwise lost connection, SteamUser will automatically attempt to reconnect. Unless you messed with the autoRelogin option, the only situation where you'd need to call logOn again is if the error event got emitted.

You're probably not checking whether your client is still connected to Steam. In that case, whenever Steam goes down (or you otherwise lose connection), your web session will expire, triggering sessionExpired, which will make your code call webLogOn, which will fail with the given error because your client is no longer connected.
You should check whether client.steamID is not null to make sure you're still connected.