He's refusing to give religious exemptions to those Catholics who have moral reservations of receiving the deathvaxx.
His refusal of "vouching for one's conscience" is just a smokescreen for:
Nope. Bp. Daly is one of the good guys.
He can’t give an exemption because this is a prudential judgment, not a question of doctrine. Use of the vaccines constitutes remote cooperation with the evil of abortion. In this case, the cooperation is very remote indeed, so that there’s no clear connection between taking the shot and encouraging future anti life actions. This isn’t casuistry; it’s sound traditional Catholic moral theology. If you have anything in your house made in China, you’re cooperating with evil much more closely. Our Lord himself acknowledged that in a fallen world, we are tangled up in evil, sometimes in ways impossible to avoid. Mary and Joseph cooperated with the Augustan tax edict, and Jesus himself did not forbid rendering unto Caesar. As much as I despise and distrust Cupich, I believe he’s on solid ground here.