As far as I know, there are no features which are restricted to the "type" of app, so game versus app should make no difference.
You can open the FB.ui({method: 'apprequests'}) Facebook Javascript dialog on any web page, the app doesn't need to be in a canvas or page tab. The wording may be confusing, but what is meant is that, when a user does accept an invitation, they will be sent to the app canvas.
This means that you will need to configure your Facebook app to have a namespace (apps.facebook.com/<namespace>
) and you will need to handle users coming from invitations at that point. What you do with them afterwards is up to you, you could redirect them to your website if you wish.
You should be aware that the Facebook Requests process is deliberately quite cryptic to avoid gaining information from users who don't want to be involved in your app. Unless you have a specific reason to track invitations, I suggest using the FB.ui({ method: 'feed'}) dialog with a target_id.
Because of this, there isn't and will not be a way in PHP to perform this. You will need to load the JS SDK onto your page and invoke the dialog, then allow the user to make their decision (whether using the 'feed' or 'apprequests' dialog).