Probably not and if someone is telling you that you can (or should) they need to straight up check themselves, because life is serious right now. Hyperemesis Gravidarum isn’t like a nasty cold. You don’t catch it, take a few days off and then head back to work pumped up on cold meds. Instead, it’s progressive. You’ll do better if you rest but even if you do, you’re still going to have it, it’s not going away for a while.
If your pregnancy is like many HG pregnancies, you’ll probably be okay to work by the third trimester, however, when you’re in the throes of this condition your priority must be rest, hydration and maintaining until your pregnancy until you’re able to stop vomiting and start gaining weight and strength.
Many women can go on short-term disability with an HG diagnosis, so talk to your OB about what steps to take to do this in your state if need be.