Do we vaccinate sick people?
Yes. Sick people are occasionally administered vaccines. But you need to keep in mind the mechanism of action of vaccines. 

Broadly speaking, there are two types of vaccines. 

Active immunization works by exposing the immune system of an individual with a pathogen (be it organism or derived proteins, etc). This type of immunization results in development of adaptive immunity in the individual, with long lasting effect; but it can days or weeks to fully develop.

On the contrary, passive immunization works by directly administering protective agents (antibody, etc). This will provide the individual with almost immediate protection; but will not lead to development of adaptive immunity, and thus will be short-lived.

You may find this wikipedia article informative.