Is There A Proper Way To Ask For Money Or Gift Cards As A Wedding Gift Without Offending Guests?

          I think you can ask that people give you these without causing offence.
It's your wedding and it's sensible to have things that are going to be of use to you. I was just arranging to sort out two gifts of money for two weddings as I read your question.
The two couples who's weddings i am attending in the next few weeks have both added money and gift cards to their list as they would like to do renovations to their new homes and would welcome any gifts towards this.

          What you could do is write a note instead of a present list explaining that , for example you have work to do on your home and instead of a conventional present would guests consider giving you cash or gift cards towards the work?

          There will be people who will ignore this an give you what they think you should have instead of what you really want but that's weddings!

          If you see people face to face before the wedding you  can try asking them if they ask what you would like or explain that you haven't issued a wedding gift list as such because you would welcome gift cards or cash. good luck.