Many Westerners are startled at the images of America and the West in contemporary Japanese advertising. Their accidental encounter by an American tourist or businessperson on a visit to Japan often evokes surprise, confusion, and misunderstanding.
I'm not sure about what "Their" in this sentence refers to. I think there are three candidates for this:
1 is like treating images like a person, so I don't feel this is the right interpretation.
American tourist or businessperson
2 and 3 are ok in the sense that they can be a proper subject of the verb "encounter", but I'm not sure why this is followed by a "by ...". Isn't this like: "His dancing by him"?