Would any order be okay with a nun having children with IVF or artificial insemination?

Pretty much it. Correct me if I am wrong, but in most Catholic orders, nuns take a vow not to get married and to remain chaste and the lawyer in me thinks that this does stay with the parameters.

What about other religions and or sects?

I can’t imagine the Catholic church going along with this. If a nun wanted to do it, they’d release her from her vows. Leaving out the legalistics of this, the reason for the vows of chastity, poverty, and obedience is (IIRC) to enable them to dedicate themselves to God and (depending on the order) the poor. That’s why women with minor children don’t become nuns.

For Catholic orders, it stays within the rules as far as not being married and remaining chaste - but according to Catholic teaching both artificial insemination and IVF are unacceptable even for married couples

CMIIW but I believe that the Catholic Church has officially come out against invetro fertilization for anyone.

What about nuns volunteering to gestate “leftover” embryos that slated for destruction or just waiting for freezer burn to set in?

Yes, you are right.

They say that in vitro fertilization is inherently evil, even if you fertilize just one egg with one sperm. But what’s even worse is that the usual IVF procedures fertilize many eggs and discard the unneeded ones. That is the same as abortion. So you have multiple evils all at once.

Sometimes the unneeded fertilized eggs are deep frozen in liquid nitrogen vats. If you visit some ultra-conservative Catholic message boards, they go around in circles about how to baptize these embryos. The challenge is that pouring water into a vat of liquid nitrogen would kill the embryos. And the official position is that even implanting these eggs into a volunteer mother in order to save their lives is an immoral act.

Catholic School Teacher Fired for Using IVF.

Nope.

My understanding is that they are not supposed to have any obligations other than to the church, which would bar their having any children by any means.

"All things considered, it needs to be recognized that the thousands of abandoned embryos represent a situation of injustice which in fact cannot be resolved. Therefore John Paul II made an “appeal to the conscience of the world’s scientific authorities and in particular to doctors, that the production of human embryos be halted, taking into account that there seems to be no morally licit solution regarding the human destiny of the thousands and thousands of ‘frozen’ embryos which are and remain the subjects of essential rights and should therefore be protected by law as human persons”

God, they need to get their heads in the modern century.

My local Catholic church’s priest adopted 2 kids about 5 or 6 years ago. He stayed a priest. I know he is a priest, not a nun, but the church allowed it.

There are orders which do not accept permanent vows; in that case, if a nun chose to not renew she’d go back to being a “civilian” and perfectly able to have kids, marry, etc without a specific release. But the very act of living in a community of grown up women is kind of incompatible with baby diapers (not with old-woman’s ones), as a baby is by definition not a grown up woman.

Adoption is a very different animal from pregnancy, and your local priest probably doesn’t live in a convent. Many orders would allow a nun or a monk (whether simultaneously a priest or not) to keep their status after adopting a relative’s children, but they would likely not be able to live in a convent while raising them, specially if the children weren’t of the same sex as the people in the convent.

This is the problem; it’s not like a minister in most other curches, where their “job” is just that - something they do outside of their family life. Instead, it’s more like the military. You are dedicated to the life 24 hours a day, it does everything and makes all decisions for you. There is no room for an “outside life”, let alone the finances and arrangements to give a nun or monk something other than their own tiny room and a seat at the community dining table. With this life, they don’t get an income to buy their own “things” - it’s all provided (which is what makes the vow of “poverty” practical; you don’t get to have your own space, buy your own little TV or whatever collection or spend your evenings all to yourself undisturbed to pursue hobbies.).

The logistics of raising a child are far outside the arangements - it’s no more likely to happen than the enlisted soldiers being able to keep a crib beside their bunk when on assignment.