How many, when did it happen and what team did it?
A quick googling says it was the Boston Red Sox with 17 runs in the 7th inning of a 1953 game against the Detroit Tigers.
The Chicago Cubs scored 18 against, presumably, a different Detroit rotation in 1883.
The most runs ever scored by BOTH teams in an inning is 19, by the Indians (13) and Red Sox (6) in the 8th inning, April 4, 1977.
What about the record for batters sent to the plate in an inning? The description for that Red Sox game says they sent 23 batters to the plate - that’s the maximum there could be for 17 runs - 17 + 3 outs + 3 stranded on base. The 18 run inning by the Cubs could theoretically have involved 24 batters sent to the plate, though I doubt if it did.
Chicago left two men on base in their 1883 18-run inning, so they also sent 23 men to the plate.
The reason for the different information here is that “modern baseball” is deemed to have started in 1900-01, owing to the wide variation in rules during the first quarter century of the National League and the 3-4 other ‘major leagues’ that were around at various times during that period.