I'm integrating PayPal Permission API to access the customer's details in PayPal, But the documentation says that adaptive payment and permissions are no longer available for new integrations. Here I have shared the Documentation link. Please suggest If any other alternative ways to integrate a permissions API
What customer details are you looking for? If it is account profile information, Connect with PayPal can be used, with approval.
But you need to be a PayPal partner to integrate a permissions API equivalent. It will be a Connect with PayPal integration with certain specific scopes used for permissions (not the documented ones)
You need approval and guidance from PayPal in order to integrate such a new permissions API service, and for information on which specific scopes correspond to the API permission you're looking for.
A way for an account to grant permissions manually (without an API or CwPP flow) is via https://www.paypal.com/apiaccess . They need to know the classic NVP/SOAP API username of the account they'll be granting permissions to in order to do so.