App reviews, troubleshooting, and recommendations
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Do you have subscription app configured already on store?
If yes above said feature can done using custom code in the theme without using an app.
Most subscription apps should have an option to tag a customer that has a active subscription. You could then add some code to hide the "Add to cart" button, unless the customer has the subscription tag.
If you don't want to use a subscription app,
You could also use a tagging app to tag a customer that purchases the one year subscription with the date of signing up and have code that only shows the "add to cart" button if the tag is within the last year