I have only been able to find instructions that involve post-processing. Is there a process where I can take a photo of the card and set that white balance and save it for my photos? Basically setting custom white balance?
That depends on camera make and model.
On my Canon 7D and 500D, it requires me to actually take a picture of neutral gray. Then I need to go into the menu, select custom WB, and select the picture that I took. From now on, when the camera's white balance setting is set to custom, the white balance from this picture will be used.
AFAIK, this process is common on DSLRs.
On my Canon Ixus 700 compact, I don't have to take a picture first. When I select custom WB, I have to point the camera at neutral gray, and then press the menu button. This white balance will be used until I change it into something else.
You could switch on he live view and set the custom white balance manually by observing color casts on your gray/white card. If you are using strobes most likely you would need to make a picture and check color casts in it.
I do not know any automated way to set the white balance from gray-card while shooting.