I'm surprised at some of the answers on here.
Yes, it is the OP's husbands body and if he doesn't want to have a vasectomy then yes, it's his right to refuse but he's also refusing to use any contraception i.e a condom because that, aparently, inconveniences him. Well isn't he lovely.
Surely, as a couple you work together to find a solution that works for both of you.
OP has had four children, she's had two c-sections and tried numerous methods of contraception, two of which were fairly invasive, all of which caused her problems of one kind or another.
When exactly is her husband to be expected, as the other half of this partnership, to step up and do his part in this relationship to make sure there are no further pregnancies?
Sterilisation is more invasive than a vasectomy, the recovery is longer and it requires a GA, which a vasectomy doesn't.
I'm not sure I could stay with someone who is so selfish that he will not take on the responsibility of something as simple as using a condom to ensure that his wife doesn't have go through yet another pregnancy and potentially another major operation if she needs another c-section.
When I was sterilised many years ago I did it because, after two miscarriages, a still birth and three babies I was absolutely determined to have no more children.
I wasn't really concerned about how my, then, husband felt about it. I did it for me so there was no question in my mind about it. But even he who was an extremely selfish man, would not have refused to use a condom if it had been necessary.