32 times by one team.
As you may have read here before, the Colorado Rockies have hit Craig Biggio with a pitch 32 times, and lead all teams in that category. It's not entirely clear that that's more times than one team has ever hit one batter - team by team splits aren't available for Hughie Jennings, Tommy Tucker or a number of others with over 150 plunks. But it's 1 more time than Ron Hunt was plunked by the Phillies, and two more times than Don Baylor was plunked by the White Sox or Rangers. The Rockies have plunked Biggio twice as many times as the Red Sox have plunked Derek Jeter (though probably less of them were intentional). And the Red Sox have plunked Jeter 1 more time than Chipper Jones has been plunked by everyone. Of the 87 players MLB.com lists as active, with over 5,000 plate appearances, one quarter of them have fewer than 32 career HBPs - less then Craig Biggio has against the Rockies, in 570 plate appearance.
Biggio has been hit by a pitch at least once in 24% of the games he's ever played against the Rockies, despite being on an 8 game plunkless streak against them. There are 3 American League teams against whom he's been hit in 33% of his games (in no more than 6 games), but the next closest NL team is Arizona. He's been hit in 16% of his games against them.
This is only the third time in his career he's played 8 straight games against Colorado without being hit - he managed to avoid being plunked in all 8 games against the Rockies in 1999, after being plunked in their final meeting of 1998, but they got him twice in his first game against them in 2000. The Rockies also failed to plunk Biggio in their first 19 meetings, after they joined the league in 1993 in the NL west with the Astros. They (Steve Reed actually) finally hit him for the first time on August 1, 1994, and have continued to do so. Frequently.
If Craig Biggio had been hit by a pitch by every team at the same rate per plate appearance that the Rockies have hit him, he'd have 683 plunks. If he'd been hit in every game at the same rate the Rockies have hit him, he'd have 730. As of the end of last season, the Rockies had 172 players pitch for them in franchise history. 10.5% of them have hit Biggio with a pitch while pitching for the Rockies, and 21.5% of them hit Biggio with a pitch at least once at some point in their career.