What is the ruling on seeking help from a relative who has a PayPal account activated with an interest-bearing credit card, whether by asking them to purchase items for me or by asking them to receive customer payments into their account to be transferred to me later?

The condition mentioned for obtaining the credit card is impermissible because it is blatant usury (riba). Your relative must fear Allah and abandon this sin. As for your dealings with him and your request for him to buy things or pay on your behalf, there is no objection to it, provided that this does not entice him or motivate him to activate his account through the forbidden credit card. If it does, then it is not permissible due to the assistance it provides in sin.