A stepparent owes a stepchild the same duty of care in their home as any child visiting. This is true in an emergency situation as well. How you are expected to behave depends upon the facts and circumstances surrounding an event, but the stepparent still cannot give legal authorization for treatment of a stepchild at a hospital or other medical facility. Only the natural parent(s) or legal guardian(s) can consent.
Some states (such as Arizona) allow the rights of the natural parents or legal guardians to be delegated to the stepparent through a Power of Attorney form, signed and notarized by a parent or guardian, giving a non-parent the right to make decisions on behalf of the minor child. It can be limited in time (from such and such a date to such and such a date), and it can be limited as to what is permitted (sign the child up for an activity, take the child to the doctor, take the child on vacation, make a specific decision). Not all places would accept such a Power of Attorney and they do not last forever. In Arizona, for example, they are valid for a maximum of 6 months. They can be renewed, but only for another 6 months. The courts want to make sure that there is not interference with the rights of parents or of the court appointed guardians.