I would definitely support the secretary in her refusal to use her personal credit card to pay these kinds of expenses.  It is the responsibility of the board to make arrangements for paying for church expenses, not her responsibility.  If the church board doesn't want to get a credit card, then THEY - the church board - can come up with a suitable alternative.   Maybe one of the church board members is willing to use their own credit card.   My guess is that at some point this will get old, and the church will decide to get a credit card for the church use. 

I think there is wisdom and value is setting up guidelines in how and when it gets used - transparency is good. 

However, I don't think this should be put on the secretary. It is not her responsibility.