Proper etiquette indicates that people send thank-you notes to people who send flowers, anyone making donations, pallbearers, people helping with the funeral arrangements and musicians performing at the service. People outside of these categories do not need a thank-you card unless someone wants to send them one.
There is no strict rule on when to send funeral thank-you cards, but it is appropriate to send them out within 2 weeks of the service. It is fine to have someone else close to the deceased write the cards if immediate family are not up to writing the cards themselves. When this occurs, the immediate family can choose to sign the cards or not.