Would I be sinful if I allowed my brother to use my card to buy a permanent tattooing machine—knowing that selling and buying it is forbidden—and he is buying it for trade and profit, even though I did not pay for it?

If the sister permits her brother to use her card to purchase a tattooing machine, while she knows his purpose, then she is sinful for assisting him in something forbidden, and she must repent. However, if she did not know, then there is no sin upon her. And she should advise her brother with wisdom to cease selling the forbidden.