Will a Board of Education sponsor a visa?
A Board that hires an ALT directly is normally the employing organisation for the immigration paperwork that goes with the appointment. Boards rarely take applicants who are not already in Japan, so in practice this usually applies to somebody changing employer rather than moving to the country.
What status is granted, and on what conditions, is decided by the Immigration Services Agency and not by the Board. Anything turning on the outcome is worth confirming with them, or with a certified administrative scrivener (行政書士), before relying on it.