Hi 4072!
I'll be happy to guide you with the How-do-I part. I will also mention certain accounts, and you can verify them or ask advice from an accountant if you're not into the accounting stuff.
Since the personal credit card is not part of their business, we can assume that the money is equity or an owner's equity money. Thus, they can record the purchase as a journal entry: a debit to the expense, and a credit to the owner's equity account.

When they pay the credit card company for the business part, they can record a check against their checking account. Then, use the owner's equity account in the Expenses section.

Usually, we put the owner's name as the payee if the reimbursement goes to the owner. But in this case, the reimbursement goes directly to the credit card company that's why we select it as the payee.
If you have other questions mind, you can always go back to this thread. Have a good day!