I've been doing some research into a Great Lakes wreck diving trip in the next year or so and my impression was that having a C-card on board with you during the charter was a Coast Guard requirement and a violation here would impact the boat operator more than the diver with the absent card. Just based on this, I'd think it's good practice for the captain to ask to see everyone's card on EVERY trip, regardless of whether you're a regular with them or a first timer or filled out all your paperwork at the office.
Yes, each diver having their c-card on them is a Coast Guard requirement. The usual charter op I go with spells this out on their website and captain requests to see my card on each trip, even though he knows me now.