I need to book and pay for my daughter's ticket to Hamburg. How do I do that?
You only need to book it as you were to book the ticket for yourself. It's the same process, but all the information on the booking will be your daughter.
However, you can use your card to pay for the entire booking. Since you're booking it through an Online Travel Agency (OTA), you won't have to provide your credit card at check-in as proof as some airlines require payment verification before or at check-in.