It would definitely be possible for male-male couples. Female-female couples are much more challenging.
Here are the embryo generation procedures for same-sex couples (the embryos afterwards are either implanted in the womb of one of the parents, a surrogate, or an artificial gestator):
Male-Male Couple → Male or female child:
This one is easy, since there's an X chromosome, a Y chromosome and plenty of sperm. I never want to type those last three words again.
Both parents provide sperm, which is centrifuged and separated by weight (the X chromosome is heavier than Y). An X sperm has its nucleus removed and inserted into an enucleated egg cell, which is then fertilized with an X or Y sperm from the other parent (they can choose or leave it to chance).
Alternatively, you could use stem cell technology to program a man's germ cells to produce eggs. The DNA is produced the same way for eggs and sperm, so men have all the necessary hardware to, with a little tweaking, produce viable human egg cells.
Female-Female Couple → Female child:
The egg cell is comparatively easy to get, just take one from the ovary surgically. The sperm cell is a different story. Unlike in the case of the male-male couple, we can't necessarily replace the nucleus of the sperm cell with the nucleus of a female egg cell. The problem is that the nucleus of an egg cell is several times larger than the sperm cell. It would be like trying to put an egg into a grape: surgically impossible.
Also, sperm cells are really fragile. When you have 1 egg and a million sperm cells, the egg has to be really durable, but it doesn't matter if a sperm cell malfunctions. Sperm cells often have defects like 2 tails or 2 heads, both of which prevent the sperm from reaching the egg. Unfortunately for Kikazaru, this means that trying to replace the nucleus of a sperm cell (a process that involves putting a huge hole in the cell membrane) will probably make the sperm non-viable.
We also can't get the woman to produce sperm by manipulating cell specialization, because the woman's germ cells (the cells that produce eggs and sperm) become inactive after birth, so adult women can't produce new eggs at all, let alone sperm.
To get around this, we would have to remove the DNA from the female and directly put it into the sperm cell. It can't be from any cell in the mother – otherwise the child would be half-cloned from the mother. The DNA has to be taken from the germline, after recombination (the process that shuffles the DNA from 46 chromosomes to produce the 23 that go into the egg or sperm). Typically, to get DNA from a cell intact, we would find cells in metaphase, when the DNA is grouped into neat chromosomes (with chromosomes, you can count how many there are under the microscope to make sure you have them all). But female human germ cells turn into eggs before birth, and stop dividing – they never enter metaphase.
So you have to take the germline DNA in interphase, when it's grouped into a structure called chromatin, which is loose, tangled and messy. It also tends to fill the space it's given, so it will be the same size as the egg nucleus. So you would grab all of the DNA, then squeeze it into the sperm, trying not to make it explode in the process. This is super messy, so you'd have to repeat it over and over, and women have a limited amount of egg cells, so you'd have a good chance of rendering the woman infertile while failing to produce a child.
A possible solution would be Artificial Gene Synthesis, also known as DNA printing. This technique allows researchers to synthesize DNA strands of any sequence they want. Currently this works by creating random DNA until the right sequence is found, so it takes longer the larger the DNA you want to create. In the future, though, we might be able to produce entire human genomes in the lab (I'm guessing Kikazaru and Protogon already can do this as part of the designer baby process).
The lab would sequence the mother's DNA (read all of the genes) and then reprint them into a germ cell donated from a man, which would then recombine the chromosomes and develop into 4 sperm cells by meiosis. Then, one of the sperm would be sucked into a pipette and forcibly shoved at the egg, which it would hopefully fertilize (if it doesn't, rinse and repeat).
Female-Female Couple → Male child:
Same challenges as above, with the addition of the Y chromosome problem. To make a boy, you need a Y chromosome. A same-sex female couple has 4 X chromosomes between them. To get this to work, you'd need a male Y chromosome donor, so a small part of the child's DNA would come from a third 'parent.' Luckily, Y chromosomes aren't really that important. Their DNA consists of 'Be a dude' + a few random genes, including the gene for hairy ears (for some reason). So the DNA of a healthy, hairless-eared male would be sequenced and would replace the X chromosome in the Artificial Gene Synthesis process described above.
Bonus: Babies with more than two polyamorous parents!
If we're running everything through sequencing and gene synthesis, we can just treat DNA as data. This means that you could have a child with yourself, using an algorithm that finds the healthiest recombination of your own DNA to avoid inbreeding (like cloning with a little bonus shuffling). You could have a kid with 10 other people, where you each contribute a couple of chromosomes, and the algorithm does the rest.
And much, much more!
You could have a child with a dead historical figure whose genes have been extracted from their hair, or with a celebrity. There would be crowds of people trying to get DNA samples of famous people to sell on the DNA black market. Give birth to a baby clone of Tom Cruise for just $5000!
Who said anything about human-only DNA? Make your baby able to photosynthesize! Make them digest cellulose! Have permanently blue hair! Take your pick.
Yeah, designer babies are pretty freaky. Once you have the tech to make a male child of female parents, you can totally play god with humans.
