A. Archer Vandegrift went to the University of Virginia in Charlottesville, VA, 1906-08; Charlottesville is where he grew up.When he retired from the Marines, he and his wife (my great-aunt) moved to Lynchburg, near where she grew up at Kenmore, in Amherst County.
After the death of my great-aunt, he married again (Kathryn) and moved back to Charlottesville.
The date in your book does coincide with when he was in college. When his wife died and he remarried, it is very likely that books in his home were disposed of.
I do have a sample of his handwriting.You may contact me at firstname.lastname@example.org and I will be glad to send you an image of an inscription from a book.