I think if a child is so upset they vomit, the only thing to do is wait. But then I am not very good at sleep training.....:-)
My DS1 used to do the vomiting if I didn't bf him, even when cuddled all the time by his dad. My DH took over doing night wakings at 16mo and amazingly, once he accepted DH as an acceptable alternative (not saying that was easy, but DH didn't leave him at all so he was cuddled all the time even while furiously upset) he quickly settled. He gradually slept for longer and longer periods. Now he is two and wakes once or twice a night but is in his own bed.
The big difference is that my DS didn't really need the food at 16mo - he definitely did at 6mo - so it was possible to stop feeding so much at night. And we realised he likes to be close to people at night so we accepted he wasn't going to self settle any time soon.
Now one of us lies down with him till he goes to sleep - it's not perfect and he still can't self settle before bed (although he is improving in the night, waking and then dropping off back to sleep sometimes). But he is never upset at bedtime and given that we both work and are exhausted, the easy route works for us. (Plus we both secretly love the cuddles when he says "mmm my mummy" or "oh daddy" as he snuggles up to go to sleep with us.)
Sorry I don't have a simple solution. DS did that 45-minute cycle of breastfeeding all night too, so I know how hideous it can be. Until the 16mo point, DS would wake eight times or so at night to feed, so I know it can be brutal.
Just wanted to say you're not alone. Those tricky vomiting-when-crying-so-much babies are very hard work. I never found a good way to deal with it except time, so now I'm having DS2, I shall see if anyone on this thread does have any great tips.