I’m not a WP API expert.
It sounds like you want to store more information than the API supports, so you will need to add an endpoint – https://developer.wordpress.org/rest-api/extending-the-rest-api/adding-custom-endpoints/
If you are doing payment processing, I assume you are using a service or plugin + service for that, so you may want your custom endpoints to interact with those DB structures that the plugin creates. (maybe you are using https://wordpress.org/plugins/stripe/ for instance?)
In addition to that, you probably want to update something else on your site, like what they purchased, or what abilities they have now that they have purchased it. So that might mean more endpoints and more DB structures.
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WordPress does not have API endpoints that save/load data (like parse or something would.) So you would need to create that by hand.