There is nothing wrong if you pay for the Hajj of your relative with a credit card provided your use of the credit card is itself permissible.

Dr. Monzer Kahf, a prominent economist and counselor states the following: It is permissible for you to pay for a relative to go for Hajj with a credit card if the following two conditions are fulfilled:

1. You only use the credit card to the extent that you can pay within the grace period so that no interest is accrued by any delay till after this period and the interest clause of the agreement is not triggered, that is, that no interest is ever generated.

2. You do not withdraw cash using the credit card because cash withdrawal is subject to interest from the day of withdrawal, that is, it does not have a grace period.

Allah Almighty knows best.