I know what you mean, it can be a very difficult balancing act even for healthy people between work, home, family, etc - and it's all much more difficult when you're ill all the time. Are you on any medications right now? If you're on medication but still vomiting daily, you may want to discuss with your doctor if there are other options available for you to try. You could also try things complimentary to medicine, like dietary changes, gentle exercise, meditation, etc. Have you had any bloodwork done lately? If you're low on things like b12, iron, vit d, etc, that could all have a big effect on how you're feeling and your energy levels.
I don't know what your finances are like, but is it possible for you to perhaps take some time off of work or cut down your hours? It sounds like you have a lot to juggle right now. Is your husband supportive, does he know how poorly you've been feeling, and does he help out with the kids and housework and stuff?
I don't have kids myself, but I've heard others on the forum say that they bond with their kids by lying in bed together and watching movies. You don't have to be super active to have quality time with your kids - surely they know that you're ill and not up to doing much, but just spending some time with them in any regard would probably be appreciated by them.