The architecture you describe is a common design pattern for a product (set of applications) hosted on the internet, where your users/partners have the option to provide an IdP or utilize your IdP for authentication. Many regulated industries require that Authorization be performed by the party providing the web applications and services. Since you will be doing authorization and adding attributes, you will need to manage all the user identities and have information provisioned for those users into your local database. Upon receipt of the authentication assertion, then it will be augmented with authorization information from your local database.
There are many on-premise Federation Solutions on the market today that perform the function you have described. I am going to focus on a SAML solution here, although there are other options for federation protocol. A couple of terms so the answer is more clear. The Identity Provider(s) will be the components that issue SAML assertions and only perform authentication, the Service Provider (SP) will be the component in your network to request/receive SAML assertions as well as augment the assertion with authorization data from your local database, and web applications to receive identity tokens, which are your applications.
Within the context of your network you will have a federation server that acts as the SP using the SAML protocol to all the desired IdPs. This component is essentially a federation hub. All of your web applications will communicate with this federation hub. An IdP Discovery service will be needed to determine where to route the SAML Request, which can be implemented within the federation hub, or in each application. My preference is to have the IdP Discovery as part of the federation hub. There are a couple of options for discovery such as using URLs or having a selection interface displayed to the user which is driven by the type of requirements and use case (workforce, business partner, customer). When your web application invokes the federation hub and associated IdP Discovery, then the SAML request will be sent to the appropriate IdP. Upon receipt of the SAML response, the assertion will be validated and subject retrieved. This is where some options come into play based on the vendor and business requirements you have for the solution. Products that act in the SP role as a federation hub that receive SAML assertions generally will have some type of plug-in interface or configuration that allows you to query for attributes from your local user database (or a directory service). Once all the data is combined, then a last mile integration occurs, either using SAML protocol, vendor protocol, or web service to the federation hub to get all the identity information. Since you have requested product that meets this use case, I have worked extensively with PingFederate by Ping Identity that will solve what you are looking to do.