An appeal for a history of VBS appeared on Religion in American History blog. My experience is that it was a summer extension of Sunday School. It used to run two weeks but is no commonly just one week. These days there is more emphasis on the “vacation” part of the name rather than the “school” part. I hope someone follows through and writes the history!
I did find on the EBSCO site “An Improved Vacation Bible School?” Christian Century, 69 no 32 Ag 6 1952, p 893. This brief article from 1952 mentions the growth of VBS as a recent development in religious education. A historian would need to verify whether this means that VBS was new at that time or whether the growth was new.