The answer to this can be found in Chapter 3 of Farewell to Manzanar. In that chapter, Jeanne Wakatsuki Houston discusses being sent to the camp. She does not give an actual date, but we can determine that her family was relocated in April of 1942.
We know that the relocation happened in 1942. One way we can know this is by knowing that President Roosevelt issued Executive Order 9066, which authorized the relocation, on February 19, 1942. This order is mentioned on p. 14 of the book. On the next page, Wakatsuki Houston mentions a time, “about a month later, when we were told to make our third and final move.” That would indicate that the relocation happened in April of 1942. We are also given other hints that it was April. On p. 16, Wakatsuki Houston says that her mother had grown up in Eastern Washington and knew that “anywhere inland in early April would be cold.”
Thus, our knowledge of history, along with these passages from the book, tells us that the Wakatsuki family was relocated to Manzanar in April of 1942.