S.W.
Only one day in all the years of child raising did we leave my step-daughter with a woman from a company like this. She was 2 years old and mostly recovered from an illness, but not able to go back to daycare yet. It worked fine, the woman was older and very capable. I don't remember the company name.
When kids are sick, it's best to have a parent home. I would check both your and your husband's company policies, but I would think between the two of you, you could stay home with a sick child, especially the first couple of years.