I have always been an organ donor on my driver license but now I am wondering as a new Catholic what the Church’s stance is on this.
In theory, when considering the issue of donations from those who are dead to those who are living, the Church doesn’t have a problem with the donation of organs from people who are already dead to save the lives of those who are alive (CCC 2296). But the important condition in such cases is that the donor must be “already dead.” Unfortunately, most organs are useless once the donor’s heart has stopped beating and so medical professionals keep the donor’s heart beating while the organs are harvested. Moral theologians question whether this procedure in fact violates this principle from the Catechism of the Catholic Church (linked above):
It is not morally acceptable if the donor or his proxy has not given explicit consent. *Moreover, it is not morally admissible to bring about the disabling mutilation or death of a human being, even in order to delay the death of other persons *(emphasis added).
I can only give you my personal opinion; but, as someone who once carried a donor card and now no longer does, I would encourage you not to carry a donor card. Discuss your wishes with an orthodox Catholic who can act for you when you are unable to state your own wishes. (For most people, this should be a spouse or someone else who can be given medical power-of-attorney.)
One alternative to carrying a donor card would be to empower your proxy to donate your corneas (which can be used after cardiac death) and then donate your body to science. (Medical schools generally will accept bodies without corneas but want everything else intact for study purposes.) Just be sure that your proxy will receive back your cremains once a medical school is done with your body and cremates it, and then arrange for a Catholic burial for you.