Actually, there is an app called contacts. It is possible you have moved it to an alternate Home screen (swipe the screen to the left and right to view them), or placed it into a folder. If you Swipr to the right from teh first home screen, the seach screen displays. Type in Contacts and see if it appears (if it is indeed in a folder, it will tell you which one).
Another approach would be to open the Settings and go to General > Reset > Reset Home Screen Layout. This will not remove any data, but place all app icons back to factor settings in terms of layout.
The Contacts app is built into iOS and cannot be deleted.