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Authenticating third party API



I am trying to create a client portal system Where I have only 10-15 customers to give service to. I created their profiles on a third party application and want to use data from the third party tool via REST api. The API allows cURL basic authentication, it checks user against a secret key.
At this moment I have php session based authentication which is buggy. I want to be able to use Auth0 to authenticate the user.
Step 1: User fills auth0 login form to login.
Step 2: System check if user has a secret key which matches the third party secret key.
Step 3: If yes then provide the user REST api access.

Is this possible?