Hey, we've got broad shoulders!
Don't beat yourself up too much. Morning sickness is hideous.
If you can manage it, how about just setting some really small targets such as ... brush hair, get dressed (even if it is into a tracksuit, or clean pj's), and get the children dressed and fed. Then I'm sure you will feel a bit better. If the sickness passes at all, then you can do a little bit more later in the day. And if it doesn't, don't worry.
(just for the record, I'm still in my pj's as are the children, the dogs are unwalked and unfed, the guinea pig unfed, and the horses are being fed by a friend this morning, so I don't exactly feel like I'm flying!!)
Do you have anyone around you who can help with the kids at all while you feeling so sick?