Some usage of "go into" allows this sentence:
The company went into bankruptcy.
Could "go into", then, be used like this:
The two teams went into stalemate.
where the context is sports?
Also, is "go into stalemate" usual?
Google N-grams does not record 'go into stalemate', 'go into a stalemate', 'went into stalemate' or 'went into a stalemate'.
'[reach] a stalemate' is more common than '[reach] stalemate'. You can use '[become] stalemated' as well as 'become a stalemate'.
Note that in chess, a stalemate means that the player whose turn it is cannot move at all. In eg football, political negotiations and war, things keep happening, although maybe slowly, and a stalemate can be broken by something extraordinary event.
You could say:
The game/match had become a stalemate
The game/match reached a stalemate
The two teams reached a stalemate