Fortunately, today's question is on a similar topic: "Do you know if the [LDS Church] has any sort of opinion/position on using and/or donating to sperm/egg banks?"
The answer, after some quick research, is yes! According to Handbook 2: Administering the Church, which, as I understand it, is a reference for bishops, stake presidents, etc. to use, the Church's policy can basically be summarized by the following short paragraph, found under the heading of "In Vitro Fertilization":
"The Church strongly discourages in vitro fertilization using semen from anyone but the husband or egg from anyone but the wife. However, this is a personal matter that ultimately must be left to the judgement of the husband and wife. Responsibility for the decision rests solely upon them."
A similar message can be found under the "Artificial Insemination" and "Semen Donation" headings, although the former adds:
"Artificial insemination of single sisters is not approved. Single sisters who deliberately refuse to follow the counsel of Church leaders in this matter are subject to Church discipline."
So there you go. Basically, the Church is opposed to sperm which is not the husband's and eggs which were not already in the wife. According to my research, if you're following the guideline of using your own sperm and eggs, the Church supports you in getting in vitro fertilization or artificial insemination.
That said, it's not altogether clear why using outside sperm or eggs is discouraged. They only mention that "responsibility for the decision rests solely upon [the parents.]" Perhaps there may be discontent when a husband feels left out because he is not the "true" father of the child, while the mother is (and vice-versa.) Perhaps the only way to know the quality of your "seed" is to have carried it around in your own body. It's hard to say.
However, the Church's position is fairly clear: use of your own sperm and eggs is strongly encouraged, while use of an outside party's reproductive material is discouraged.
While we're in this manual, I would also like to note the Church's policies on a couple of other, related topics:
Abortion:
This section in the manual is slightly longer than the others. It is very illuminating, however. Basically, it says there are three situations where abortion would be acceptable:
"1. Pregnancy resulted from forcible rape or incest.
2. A competent physician determines that the life or health of the mother is in serious jeopardy.
3. A competent physician determines that the fetus has severe defects that will not allow the baby to survive beyond birth."
It states, however, that "these exceptions do not justify abortion automatically. Abortion is a most serious matter and should be considered only after the persons responsible have consulted with their bishops and received divine confirmation through prayer."
My husband and I, had we chosen to abort, would have fallen into the third category, had we chosen to abort our daughter due to her birth defects. Abortion is such a touchy subject, and most people are either taught "Yes, it's okay" or "You'll be scorned if you do." It's really not that simple, even within the Church.
Birth Control:
"It is the privilege of married couples who are able to bear children to provide mortal bodies for the spirit children of God, whom they are then responsible to nurture and rear. The decision as to how many children to have and when to have them is extremely intimate and private and should be left between the couple and the Lord. Church members should not judge one another in this matter.
Married couples should also understand that sexual relations within marriage are divinely approved not only for the purpose of procreation, but also as a way of expressing love and strengthening emotional and spiritual bonds between husband and wife."
Surrogate Motherhood:
"The Church strongly discourages surrogate motherhood"
Surgical Sterilization (Including Vasectomy):
"The Church strongly discourages surgical sterilization as an elective form of birth control. Surgical sterilization should be considered only if (1) medical conditions seriously jeopardize life or health or (2) birth defects or serious trauma have rendered a person mentally incompetent and not responsible for his or her actions. Such conditions must be determined by competent medical judgement and in accordance with law. Even then, the persons responsible for this decision should consult with each other and with their bishop and should receive divine confirmation of their decision through prayer."
There are quite a few other topics to explore, but I found these to be the most applicable. If you still have questions, I'd be glad to address them, or you can read the handbook for yourself on LDS.ORG. As I mentioned, the handbooks are meant to guide bishops in their duties. As such, they hold the "official" answers to many burning questions an individual might have about the Church. I encourage you to search them sometime. You might be surprised to see what's there!
Thanks for the question! It was refreshing to write again. Please don't hesitate to ask me some more!
Lora