Can someone be involved in a lesbian or gay relationship and still be a faithful Catholic?
Certainly yes, not as a matter of public Church teaching but as a matter of conscience, as a matter of personal application of the whole of Catholic teaching to their particular case.
In 1975, the Vatican published a Declaration on Certain Questions Concerning Sexual Ethics. One of those questions was homosexuality. A principal author of that document was Fr. Jan Visser, C.Ss.R. In an interview in L’Europa, January 30, 1976, he said, “When one is dealing with people who are so deeply homosexual that they will be in serious personal and perhaps social trouble unless they attain a steady partnership within their homosexual lives, one can recommend them to seek such a partnership, and one accepts this relationship as the best they can do in their present situation.” One of the very men who formulated the Vatican teaching that homogenital acts are wrong allows that in certain cases one may not only permit but even recommend a homosexual relationship.
Similarly, speaking about Catholics who dissent on Church teaching about contraception, the Canadian bishops wrote in 1968, “Since they are not denying any point of divine and Catholic faith nor rejecting the teaching authority of the Church, these Catholics should not be considered nor consider themselves cut off from the body of the faithful.”