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Let us assume a hypothetical situation in an MLB game. A pitcher gets to a 2-2 count against the batter. Then the pitcher leaves the game - perhaps he sustains an injury while delivering a pitch, or is ejected by an umpire. In this case, the offense is allowed to substitute a new pitcher during the at-bat because the old pitcher left due to injury or ejection.

Which pitcher is credited with the result of the at-bat? Does it matter what the count is? Clearly if the batter hits a home run that is the responsibility of the new pitcher, but if the batter walks after going 3-0, would the new pitcher be credited with the walk after the first pitcher put him in the hole? Same with 0-2 and getting a strikeout.

What do the official MLB rules say about this? Section 5.10 covers substitutions including some pitcher-specific rules, but I cannot find anything about during an at-bat. Section 9.16 is pretty thorough with regards to swapping pitchers between batters in ERA calculations, but I have been unable to find anything about pitching statistics where a single batter faces two pitchers during a single at-bat.

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You needed to look in rule 10.16(h). Rule 10 deals with the official scoring of MLB games. So based on the official rules at mlb.com:

(h) A relief pitcher shall not be held accountable when the first batter to
whom he pitches reaches first base on four called balls if such batter has a decided advantage in the ball and strike count when pitchers are changed.

(1) If, when pitchers are changed, the count is 2 balls, no strike, 2 balls, 1 strike, 3 balls, no strike, 3 balls, 1 strike, 3 balls, 2 strikes, and the batter gets a base on balls, the official scorer shall charge that batter and the base on balls to the preceding pitcher, not to the relief pitcher. (2) Any other action by such batter, such as reaching base on a hit, an error, a fielder’s choice, a force-out, or being touched by a pitched ball, shall cause such a batter to be charged to the relief pitcher. Rule 10.16(h) Comment: The provisions of Rule 10.16(h)(2) shall not be construed as affecting or conflicting with the provisions of Rule 10.16(g).

(3) If, when pitchers are changed, the count is 2 balls, 2 strikes, 1 ball, 2 strikes, 1 ball, 1 strike, 1 ball, no strike, no ball, 2 strikes, no ball, 1 strike, the official scorer shall charge that batter and the actions of that batter to the relief pitcher.

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