Log into external site from within wordpress?

Hello, Is it possible to log into an external web site from within a wordpress site? I know it’s a strange request, but here’s the low-down.

I working with a client who would like to re-vamp the front end of their site with something wordpress based. They have an existing php based site that includes a log-in interface for customers. Once logged in will display a bunch of data based on their specific user account. They do not want to change this system at all.

Basically what I’m hoping to do is put their existing login fields onto the new wordpress based front end. When a user puts in their information, it will redirect them to the existing customer interface already logged in.. does that make sense? any thoughts or advice?

They do NOT want to move their existing customer system/data into wordpress. It’s a totally custom built thing specifically made for their business and they don’t want to mess with it.. They just want to redesign the front end that the public sees..