No!  You don't need wedding rings to get married.  A couple is not legally required to exchange rings during a wedding ceremony and it is up to the two of you to decide if you want to wear them during your marriage. 

This question is usually asked when a couple ordered their rings by mail and they haven't been delivered yet,  but some partners choose not to wear a wedding ring.

The first time a couple told me only the bride would be wearing a wedding band I was surprised.  They went on to explain that the groom would not wear the ring due to his job and his father never wore a ring either.  They decided they had other things to spend money on and I agreed - it's logical.  

Since that time, I've met many couples choosing not to wear rings.  Usually the groom is an electrician, machinist, or does work with their hands.  Some couples choose tattoos!