Login to external website from WordPress Blog

Does anyone have an idea to realize an own login window on WordPress for the login procedure for an other application on an other website ?
Background: I have access to an online application with white label option (own logo and own URL) and are enabled to share access to this application for a larger nunmber of users managed (only) by me. Now I want to give access by using an own login window on the own WordPress blog, connected to the external application website. iFrame is not the solution for my problem, because then the original login window appears on my WordPress blog. Maybe interesting that the application owner shares API-access (but unfortunately no App or Plugin).
As far as I see there is no “ready to go” solution available … or don’t I see the trees in the forest ?
Thanks in advance !