I am working on a WPF app (written in C#) that searches and serves data from the active directory. Searching for a user will fetch all phone numbers for the user and display on the app window.
How can I open the outlook contact card by the click of some icon or the username or something?

The Outlook Contact Card is part of Microsoft Outlook and (as far as I know) not a public control, so you can not simply ‘open’ it unless you are creating an Outlook add-in.
Using the Active Directory’s information you could recreate it.