Theoretically. I'm not familiar with egg donation but you need to fit a lot of criteria to be able to do it and it's very hard to find a doctor that will assist you through the process, of any kind of elective sterilisation, if you're not looking for a vasectomy specifically. Especially if you're under 35ish and haven't already had multiple children.
Edit to add that people with penises do get refused vasectomies, but to my knowledge it's nowhere on the scale for people with uteruses getting refused equivalent procedures.
The entire furore regarding sterilization is nuts. We looked into it, realized that it would be much, much easier for me to get sterilized than my partner (more reliable and cheaper, as they refuse to do actual sterilization on women on the NHS, just going with the worst method because it's easier to undo), and we were amazed at how easy it was, particularly with us both being under 30. Probably just the doctor we ended up seeing, but no "are you SURE?" or "But you might break up!" Or "You'll change your minds!!", just a check that our reasons were sensible from the first doctor, then the standard spiel about it being irreversible from the actual doctor at the clinic. The latter of which he had to go through with me both times I was in, despite doing it the first time
On the flip side, I've got a friend with severe (genetic based, afaik) Arthritis who can't get a Vasectomy, though I don't know if they just said the wrong things to their doctor or what. Quite often, it makes me glad that I've got some parts of life on easymode due to having a penis (though, of course, it's only some), though I then get annoyed at how crappy it is that others don't get similar treatment.
But it's MUCH healthier for women to be perpetually on medication with a list of side effects longer than most peoples' arms - half of which, such as MH ones, don't get mentioned at all, even when people on said medication have MH illnesses - than to lose out on even a miniscule chance of having kids, even if it kills them. The entire culture around contraception, sterilization, and having babies needs to flipping well change. Would any other medication with such a list of side effects as the pill get through any kind of screening? I severely doubt it, but it's a given that the majority of women will be on one of the many types of hormonal contraception at some point, and many will be on it for, what, 30 years? More? Even disregarding the contraception issue, people are put on it for regulating periods to some extent, and people are criminally underinformed regarding the side effects. Then we get to the fact that many NHS trusts don't offer any female sterilization at all now. If they do, it's stupid clips, not actual sterilization. "Not cost effective" - but it's a damn sight cheaper than even one kid's child maintenance for sixteen years!