Click to enlarge
What is known is that armies would in fact throw and drop rocks on enemies as a defensive and offensive tactic when protecting a castle under siege.
The architecture of a castle allowed for it, and it was not hard to imagine hauling stones up to a wall to drop of attackers.
On a battle theater through the exhumation of skeletons that verified written record of the time, it was discovered that a lot of wounds were caused by stones, spears, arrows and firearms. However throwing of rocks by hand, while dangerous, is not notable as a dependably effective method of attack; at least not as effective as the simply crafted sling.
While evidence of injury and death by stones are found and believed to be used by a mobile infantry to elicit attack, the methods are almost completely unknown. Along with record of injury, there is also documentation by several priests, records kept at temples and other accounts in documents spanning from the Sengoku era. Even then very little is mentioned, other than a note with sparse details.
Takeda's army even had "stone throwers" (投石) who practiced something they termed "ashizuri" or stomping (足摺). This was written about in the exploits detailed in the "Mikawa Monogatari" by Tadataka (notable as one of the books that contributed heavily to Nitobe's "Bushido" decades later).
It has been speculated that while this might be a legend, it also might be based in fact regarding tosekigijutsu.
There has been and is a healthy dose of speculation as to what these accounts mean, because legends tend to misrepresent reality and facts give way to fancy. As time rolled on into the era of peace the practice either was just left to the wayside and forgotten about, or relatively obscure enough to not have been mentioned to any extent. Romanticism in heroic exploits replaced the hard learned lessons on the battlefield, which were subjugated into distant history.
There are depictions of the use of the sling in art stemming from the scrolls emperor Goshirakawa gave to Mitsunaga Tokiwa in the 12th century.
In the 9th scroll of the "Nenchu Gyoji Emaki" (年中行事絵巻その九) it clearly shows the use of slings in a re-enactment of a battle during a spring festival. This practice, at the time that the scroll was written, was about 300 years old and was said to mark the Otennmon affair in 866. The scroll series were written up along with a historical account called the "Ban Dainagon Ekotoba", by the same artist at about the same time and was said to have been a companion to it. (https://en.wikipedia.org/wiki/Ban_Dainagon_Ekotoba).
(The full document of the Nenchu Gyoji Emaki can be viewed here at http://edb.kulib.kyoto-u.ac.jp/exhibit/b35/image/09/b35s0136.html).
There are other depictions and records of slings being used with rocks in Japan, especially along the Watarase river in Gunma prefecture and in the festivals of Ibaraki.
The now dead tradition in Gunma was called "Watarase gawa no ishigassen”. It was an ancient ceremony that eventually stopped because of serious injury to the participants. Prior to the banning, it is thought that the event preceded any official ceremony and formal record and it was linked closely to the local shrines of the area. In the records of the festival's practice it is written that the participants were throwing stones not only by hand, but with slings. During the 17th century, prior to the festival's banning, depictions of the event pop up along with descriptions, showing the use of slings and hand throwing.
Another festival in Ibaraki was performed a similar way. The legend goes that the god of the Kashima shrine has a love affair with the god of the Hachiman shrine in Kyoto. During the years when the couple were not fighting, pestilence and disease is said to spread across the land. So, according to this legend, the people of the neighboring towns and villages would gather together for an ishigassen as an act of protection against illness. These events were held in May of every year, generally corresponding with "boys day" (now known as "Children’s Day" on May 5th) and continued up until 1940. Again, the use of the sling is recorded there as well.
So, while there is no particular school associated with throwing rocks at the enemy (especially by sling), whether it was because it was deemed uncouth of the warrior and un-favored and forgotten by design, it ceased to be practiced because of function, or it was simply not written down well enough to be of concern, there seems to be hints of around the fringes within culture.