Full disclosure: My DD was born when I was 45. No drugs, no IVF, just nature taking its course. My 2 children are 10 years apart, and I had several miscarriages in the years between. It hurts no matter how many times it happens, and no matter what I might think of the Duggars, I still feel badly for their loss.
The odds are quite high that the child the Duggars lost had a trisomy. At 45, her eggs are a lot less viable now, and as someone who has BTDT, I can attest that doctors know that the odds of miscarriage go way up for chromosonal irregularities as eggs age. In my second to last pregnancy I did end up with a life-threatening condition, but it was not due to my own health, but that of the child, caused by a fatal trisomy. It is a fairly likely bet that the Duggars would not undergo early invasive testing, given their views.
I think that the reason one's life is in danger has great bearing on decisions like these. My personal feeling is that when her own medical issues mean that pregnancy has begun to be a predictably major life risk for Mom, then it is irresponsible for a woman with existing children to deliberately risk it again. In this case I'm sure that the perspective is that the extended family would care for the younger children if their mother died, but a parent is a parent, and the loss of one in childhood is devastating. As Michelle has a history of pre-eclampsia, I think that she falls into this category, and for the good of her family it is time to stop now.
As to a vasectomy, it won't happen. He won't do it because IF she dies, then he will want to remarry within the Quiverfull movement, and no woman within the movement is going to want to marry a man who has been sterilized.