What kind of approach would a Modernist author use to tell a story about library books that have stories of personal events in the life of the reader?

Story within a story.

Explanation:

The story within a story is a literary technique used by the writer to write a piece of narration in which a narrator narrates another story or tale. The story within a story technique creates a set of multi-stories.

In the given case, the modernist author would use the approach of the story within a story. As it consists of a frame story with embedded stories. The frame story is the one in which the main story takes place ad which prepares the room to tell other stories. And embedded stories are those stories that are told within the framing narrative.

The scenario that has been given consists of a frame story about library books, that have stories of personal occurring of their own, embedded stories.

Thus the correct answer is the story within a story technique.