Cognito authentication with no keys

It sounds like they may be using a Public client in Amazon Cognito, see App Client Types, in which case you can initiate the authorization flow with just the client ID.

The details of implementing authorization code flow can be found in the Open ID Connect (OIDC) Authorization Code Flow documentation. Typically you would use an (OIDC) library to handle the authentication flow, I don't have a personal recommendation for PHP.

One thing to note - typically you would use a Confidential Client for a server side application like PHP. Is it possible to ask them to create you a Confidential client instead of a Public client? In that case they would provide you with the same information, but also provide a client secret which you would add to the request in the auth flow.