Implementing a 3rd party widget with Okta SAML SSO

Suppose our company, Company A, has an application that our clients, Company B currently use. They use Okta for authorization via SAML SSO, and it is already all setup so when they login to Okta, they can access their user and stuff in our Company A application, as well as their own internal resources in their own Company B application.

Now we are trying to build widgets so that they can embed access to our application within their own Company B application.

What I am trying to figure out: assuming the user has authenticated with Okta, and authorized into their Company B application, how can I get the widget to authorize with our own application without causing a bunch of redirects or whatnot - basically if you’re already authenticated with Okta, to silently, somehow, get the SAMLResponse from Okta and send it to our server to get our authentication token.