The term did not originate with Marvin Gaye but rather 100 years earlier.
In 1859, Colonel Bernard Bee was the first person in America to build a telegraph line and soon others followed. The wires were originally tied tightly to tree branches but lack of regular upkeep meant that the lines fell to the ground in loops.
During the Civil War, the militia used telegraph lines to transmit important information with regards to their whereabouts and what they had accomplished. Unfortunately, with so many people having access to the telegraph lines, oftentimes the information was difficult to understand or conflicted with other information about the same regiment.
It didn’t take long before the term “heard it through the grapevine” came to mean the sharing of information that had no definite source, was generally false and was most likely to be a widespread rumour and therefore false.