I am entitled to 3-months of free AppleTV but can find no way to access it online.

No, that wouldn't count. It would only be if you previously purchased any other Apple device and activated this offer, or if you belong to a family group and someone else had used it.

The only other thing would be dependent on when you purchased the hardware, as the offer is only valid for 90 days. However, you would still be entitled to the 7 days once that expires.

If you aren't sure then I would just contact Apple and they will be able to tell you.

The offer is tied to the purchase of new hardware, and needs to be activated within the TV app on that device. Once you setup that device and sign in with your Apple ID you can go into the Apple TV app and tap the banner to enjoy 3 months free (which will start an extended trial and convert to a paid subscription once that period ends).

If you previously subscribed to the service or activated another offer then you would not be eligible to receive another.

You are always free to contact Apple (which may be better than going to the store, but still an option) but if those circumstances were determined it would be under their discretion as to making any exception.

You're welcome. Have a good night and a happy new year!