Start here from the PingFederate .NET Integration Kit User Guide -
https://documentation.pingidentity.com/display/NETIK251/Processing+Overview
For SP-Init SSO, your application would redirect to your (SP) PingFederate Server which generates an AuthnRequest to the IDP. The IDP Authenticates the user and sends a Response back to your PF (SP). Your SP validates the message and maps the user's info into the SP OpenToken Adapter. The OTK Adapter will then generate a OTK and send it via the browser to your application to validate/consume using our .NET API for OpenToken.
Alternatively, you could check out PingOne Application Provider Services (APS) which is Ping's OnDemand solution. No installed software and a RESTful API to integrate your .NET App. However, not sure if your application is connecting to 2 different customer IDPs or 2 internal IDPs.
HTH, Ian