I'm almost 36, sometimes even I can't make it to the loo. That's why they invented puke buckets.
And yet, as the OP has already stated, her DSD has been given multiple sick buckets but "can't be sick with her eyes open" and so just shuts her eyes and throws up wherever she is regardless.
Nobody is infallible and everyone gets caught short occasionally, but at what point does missing the sick bowl/towels every time she is sick; removing mattress protectors so her mattress gets ruined because "it rustles"; making no attempt to clean herself up and instead coming to someone else to do it; soiling floors and furnishings in various rooms and disturbing other family members (who are recovering from the same D&V bug themselves) even when she hasn't been sick stop being written off as "one off accidents" and instead become a pattern of behaviour?
This child may be young but she is not a baby. My DD is 8 and she manages to throw up in a receptacle and not all over the bed/floor because even if she shuts her eyes she aims first. She also is capable of getting herself to the bathroom if she has an accident and stripping out of any soiled clothing and into the shower without getting faeces on towels, dressing gowns, bathroom floor etc. Do I expect her to deal with it all herself while I sit doing nothing? Of course not, but I also expect her not to make my job of cleaning up any harder when she can help it.
Throughout all of this, who is the one looking after DSD?
Her mother? No, she knowingly sent her into a household recovering from D&V because she had to work
Her father? No, he missed having the D&V bug as he was away when the OP dealt with being sick (and the other children being ill) and has washed his hands of any kind of parental responsibility because it makes him feel sick if he has to clear up his own daughter's bodily fluids or be involved in the process, but he does feel able enough to decide she can eat crap and do what she likes and must not be treated like she's actually ill.
DSD herself? No, she does whatever she wants, wherever she wants and makes no attempt to help deal with the mess she creates.
No, it's none of those people, instead the only person looking after this child is the OP who just happens to be a step-parent who seems to be deemed responsible enough for her DSD to clear up after her - despite being heavily pregnant, dealing with her own children who are also recovering from the same D&V bug, recovering from the bug herself - and yet is not responsible enough for her DSD to be allowed a say in whether or not she comes into a sick house or what the appropriate food/drink/activities are.