Ugh this happens to us too every now and again even at three months! Having done it all this time (ok, not long, but it feels like forever lol), here's what I've found:
In the beginning you produce a LOT of milk - far too much actually ... because of this, a new baby is often full before the milk has had a chance to become really creamy and high in fat (this happens gradually, with the milk towards the end of an average feed being much fattier than that at the beginning). In itself this isn't a problem, as all milk contains some fat and will nourish your baby just fine until production settles down. However, the thirst-quenching, watery stuff that starts the feed is high in lactose and babies can't process it too well to begin with, meaning they often get gassy during digestion. Could this be happening with you? If you're producing an extreme amount of milk, you'll know about it lol.
As for the vomiting, it's probably just a lot of spit up, though it comes out with quite some force sometimes. This is probably caused by the baby taking in a lot of milk quickly along with air - the bubbles trapped in the stomach have to come out somehow and they usually bring all the milk on top with them. This is probably caused by a fast and strong let-down that the baby is struggling to tolerate, and should also calm down as milk production regulates.
The following things may help: taking the baby off the breast during the initial let-down and allowing it to subside before continuing (babies swallow the most air during let-down when the flow is fast as they tend to gulp!), burping the baby frequently during a feed (bit of a bore but can really help) and always burping afterwards, and not offering the baby the other breast unless you are sure that she has finished the first. This should allow her to get more fatty milk from the breast. If the gas is really bad, you could try a product like Infacol which joins all the gas bubbles into one bigger bubble, making it easier to bring up.
Rest assured that everything you describe sounds perfectly normal and will pass in time. You are doing a good job - it's a rocky road and can be tough at times but it's worth it.