I am going to make a couple of assumptions about your application, mainly that it is .NET application hosted on IIS. This type of application integrates Windows Identity Foundation (WIF) using tags in the web.config and then reads the authenticated users identity information via the Claims Identity object passed to your application. In this case your application is referred to as the Service Provider (SP).
If your customer is using PingFederate, the integration is straightforward in that a WS-Federation SP Connection would be configured on their server referred to as the Identity Provider (IdP). If your customer is using PingOne, then the integration will be tricky.
The information to exchange for PingFederate is driven by the configuration in the configuration in web.config. You need to configure the thumbprint which is based on the digital signing certificate of the WS-Federation response containing the SAMLv1.1 token. Your customer will be able to provide the thumbprint value. You will also need to configure the federated authentication URL, issuer, and realm, which is the information about PingFederate IdP server. The issuer is the URL for your SP application to redirect to the PingFederate IdP server, along with the realm which equates to the PingFederate SP Connection. Be sure that you configure the audienceUris to be the same value as the realm. The PingFederate administrator will need to know your Service URL endpoint which is your SP application endpoint to receive the WS-Federation response with the SAMLv1.1 token.