The Church’s teaching regarding artificial reproductive technologies (ART) is not categorically a no. However, as far as I know, we have yet come up with a method that does not raise even more problems than it solves.
Before I list the main concerns, which admittedly probably not exhaustive as I’m not an expert in bioethics, we must understand that the Church’s issues with the varieties of ARTs are not because they are artificial. That’s not a problem at all. The problems all have to do with the dignity of life, especially that of the children.
[I know the following is not what you are asking, but for the sake of other readers, let me first explain the general issues with existing ARTs.]
First major problem is with how the embryos are treated. Most of the current methods do not create one embryo, but multiple embryos, since the failure rate is quite high, both in insemination and in survival. Doctors just want to maximize the chances given the cost of the process. But every conceived embryo is your child, is a human life, how do we choose which one to grow to full term? Often parents just want one child at a time, so the rest of the embryos are frozen or even jettisoned. Or if one is to be chosen, which one? Eugenics are often introduced here, or biochemical or other processes are employed to ensure the one chosen is strong enough to grow, some of such processes often destroy or weaken some of the embryos. These are all human lives, and are your children. No one would agree to subjecting their little newborn to be processed or however temporarily frozen in these manners; and no one therefore should consider such processes for embryos to be moral. Every embryo should be accepted and respected as any other born human person.
Second problem: how the sperm is obtained. The Church maintains that children has the right to be born out of love. The conjugal act within marriage is therefore not only institutional, but a participation in the procreative love of God, from the marital commitment to the conjugal act to conception. However, very often, in ARTs, sperm has to be obtained outside of conjugal act, or at least intervened not merely for collecting it, but for testing it, since more than half of the time failure to conceive is caused by low sperm counts or poor sperm health. Again, high cost of ARTs simply prohibits doctors not to make the best chances out of each trial.
The result is that doctors simply will get quite a few eggs from the wife, and as much sperm sample they can from the husband, make as many embryos as they can, screen them biochemically, pick one that they deem most healthy to grow and freeze the rest that survive. After all, if the natural one egg and sperm meeting up in the uterus is a workable solution, there wouldn’t be as much a need to try ARTs to begin with.
Regarding what you suggested, collecting the sperm from the conjugal act, performing insemination artificially, then injecting an embryo back to the wife to carry to term, I’m afraid no doctor would use only one egg or create only one embryo. And no doctor would not screen and test the embryo as mentioned above, which as stated, is not moral.
Lastly, let us also remember, while a woman’s urges to be mother is perhaps one of the strongest human desires, being able to conceive is a privilege and a blessing, and is not a right. Life is given to us, we cannot demand it. And not being able to conceive is not because we are less blessed. God merely has a different vocation for each. Some are called to raise physical children. Some are needed to devote more time to raise spiritual children. Yet some are also called to be fostered parents or adoption parents. God has a calling for each. We must learn to ask Him and listen, to first receive His Plan.
And I pray: May the Lord bless you and heal you from infertility. And yet, Lord, ultimately, not our will, but your will. May You be glorified.