OP, most women I know did not vomit during labour. It is by no means essential to.
I often feel nauseous before and during my period, not because of the pain but because of the proximity of uterus and bowel I suppose - having a retroverted uterus can I am told exacerbate this.
I also had HG during all three pregnancies. But I wasn't sick in any of my labours.
In the first one I had a heavy duty epidural and I was sick about 12 hours later - not that I had anything to bring up, and I didn't feel nauseous exactly, but I think it was side effects from the epidural or something like that (maybe the local actually, I don't know - the one they give you for stitches? as I react badly to codeine) and I kept retching. It stopped when they gave me some drugs through my cannula.
The worst thing was I couldn't feel my legs even then and it took me about 24 hours to be able to stand up, so I couldn't go to ds who was crying for a feed and no one came to help so he went back to sleep. Awful.
Second time I had a homebirth and wasn't sick but during transition I felt like I might be, purely from the pressure which was both downwards and upwards if that makes sense - it was like a mechanical thing though, caused by contractions and no nausea involved. It passed very quickly. That birth really hurt though, no drugs at all.
Third one I had another epidural but much lighter dose I think and I only felt nauseous later, I hadn't eaten much, and couldn't bring myself to, and it was that kind of vicious circle situation. But the labour itself did not make me sick.
I hope that your labour goes as well as possible, and even if you are sick it won't be the most important thing on your mind.