Has it ever happened that a player in the MLB ever won the AL or NL MVP award but didn't get elected as an All-Star that year? If so then how often does it happen?
If they didn't go to the game because of an injury or they just didn't want to that doesn't count, it has to be that they didn't get elected because the voters didn't think that they are as good as the other options.
Has an MLB player ever won an MVP award without going to that year's All-Star game?
Yes.
Since the 1994 strike-shortened season (inclusive), the following have accomplished this feat:
how often does it happen?
Statistically, 4 out of 52 MVPs between 1994-2018 have not made the All-Star Game during their respective MVP season. This comes out to 7.7%. Suffice to say, eventual MVPs infrequently do not get selected to that season's All-Star Game.
