No. Neither Pfizer, Moderna, or any of the vaccines currently in development in the United States, contain the live virus. Therefore, you cannot get COVID-19 from the vaccines.
The point of the vaccines is to teach our immune systems to recognize and fight the virus, so that we can build immunity. This is done by introducing the spike protein to the immune system, which induces an immune response. This process may cause symptoms to occur, such as fever, chills or fatigue. Symptoms after receiving the COVID-19 vaccine are normal; they’re simply a sign that your immune system is starting to build protection against the virus.