What are the tax consequences of my father and step-mother paying my step-sister to be their caregiver without giving her a 1099?

It would likely depend on how much they are paying her. People are allowed a certain amount of "gifts" per person, per year, so if they are giving her a gift, so to speak, for helping out, and it's under the amount allowed, it may be okay. If she is truly being paid to work, then it's likely taxable. You need to check with a tax expert to be certain.
Carol