The record is twelve straight at bats with a hit. It’s been done three times.
The first was the Chicago Cubs’ Johnny Kling in 1902.
The second was the Boston Red Sox’ Michael “Pinky” Higgins in 1938.
The most recent occurrence was by the Boston Red Sox’ Walt Dropo. Dropo hit 5 straight singles on July 14, and 4 more in the first game of a double header on the 15th. In the second game he added a triple, single, and double before popping out to end the streak. That streak and the 1950 rookie of the year were the big highlights of his career. My first recollection of him was seeing a picture of one of his baseball cards in book I had as a child, and that his name amused me. The team name on that card, Cincinnati Redlegs, because Reds seemed too communist at the time, also amused me. What can I say? I’m easily amused.