If Dogtor is not what the page said it would be, reply to your receipt email and the $9 goes back to the card you paid with. No form, no questions and no "let me offer you a discount instead".
How to ask
Reply to your Lemon Squeezy receipt or email nik@getdogtor.com from the address you paid with. That is the whole process. You do not have to give a reason.
Refunds are issued by Lemon Squeezy, the merchant of record, back to the original payment method. Their side usually takes a few working days and how quickly it appears after that is up to your bank.
If Dogtor is never released
Then the refund is automatic. You are not holding a ticket to something that is not coming and you will not have to chase anyone for it.
Your statutory rights
If you are a consumer in the EU or the UK, you have a 14 day right to withdraw from a distance purchase. Because the app is delivered later rather than immediately, that right is not waived at checkout and it runs alongside everything above. Where no delivery period has been agreed, delivery is due within 30 days and if it is not met you may cancel and be refunded.
This policy does not take away any of that. It is deliberately more generous than the law requires.
What is not refundable
Nothing, so far. There is one product, it costs $9 and it is a single payment. There is no subscription to cancel and no partial period to argue about.