The short answer is that "tornado season" in the United States is from March to June. This is when most tornadoes occur.
However, it should be noted that tornadoes have been recorded in every month of the year in the United States. So tornado season may come in the spring, but being prepared for a tornado or other severe weather is important 365 days a year.
Make your shelter-in-place plans and be sure to have an emergency kit to handle the aftermath should you end up in the path of a tornado.