I hope asking for clarification is okay as a top-level response.
Do you mean a person or small group of people who didn't control a state and weren't acting on behalf of one?
Perhaps something like The Ems Dispatch would be a good candidate for this sort of thing.
Bismarck was very much in control of Prussian relations at the time, so not so much in the shadows, but his edits to the transcript were designed almost entirely to provoke the Franco-Prussian war, which would enable the unification of Germany.
The idea was to remove the North-South divide from German nationalism, by being attacked by a third party, in this case France. This, following Prussia's well-prepared and executed defeat of France, would provide a cause for total German unity, given the previous crushing defeat of Austria. The South Germans needed a protector from such an aggressive France, and Prussia could then step in and claim this role, uniting them into the German Empire.
Your question seems to imply "on purpose" because the assassination of Franz Ferdinand is generally agreed upon to have started world war 1 though, of course, Europe was primed for war structurally prior to the shooting.
Absent that implied caveat, certainly that event would fit the bill.
There are many wonderful sources on the Black Hand and the assassination, none of which are in front of me right now but if you'd like a more detailed description of events I will try to scrape something together
A situation similar to what you describe is one part of the World War I origin story.
The war was precipitated by the assassination of the Austrian Archduke Franz Ferdinand. His assassination was planned and carried out by a secret society within the Serbian military corps known as "Unification or Death" or alternately "The Black Hand."
The Black Hand were ultra-nationalists whose aim was to reunite all Serbs under a single national state. They were led by Dragutin Dimitrijević AKA Apis, who was a high ranking military officer who had participated in the earlier assassination of the Serbian Monarchy in 1903.
So - that fits your criteria fairly well. However, there are a few caveats in order.
Most importantly, while the assassination of Franz Ferdinand was a precipitating event, it was hardly a sufficient condition for war. In another historical context, the assassination might have only resulted in a small regional war, or no war at all. It was only due to the very particular diplomatic conditions of the period and the decisions made by key diplomats across the major powers that the assassination led to war.
Secondly, it is debatable whether the primary motive behind The Black Hand's actions was to start a major international war. As I mentioned earlier, they were ultra-nationalists and were working towards realizing a unified Serbian state. Some of them likely believed that a war with Austria-Hungary was a necessary step toward that goal. However, it is unlikely any of them predicted that the assassination would start a war between all of the European powers.
A recent book that is excellent on the origins of World War I that you might find interesting is Christopher Clark's The Sleepwalkers. My account above borrows heavily from his book.
Hopefully not too dissimilar a question, but has something similar happened with single companies? It seems a popular position to say that a banana company was a driving force behind american actions in Latin America or individual mining operations the force behind american policy regarding African military action, is there anything concrete surrounding these claims?
Perhaps you'd be interested in Richard Neville, the Earl of Warwick? During the War of the Roses, his direct actions were responsible for putting two kings on the throne of England, leading to him being dubbed as the "Kingmaker." He fought on the side of the Yorks against the Lancansters, engineering some remarkable political moves through military battle and later, political intrigue. For a single entity, he alone was a tremendous influence on the course of English history, which I think is a remarkable thing, and might be what you're looking for (I wasn't sure based on your question).
William Walker and a group of men known as "Los Filibusteros" conducted private military expeditions into Latin America in an attempt to establish English-speaking colonies that he could use as a source of slave labor.
He briefly conquered part of Baja California and established La Paz as the capital of the New Republic of California before being forced to retreat back to the US by the Mexican government.
Shortly thereafter, Nicaragua erupted into a civil war. Walker established a contract with Castellon, the leader of the Democratic party, and gathered 100 or so military men and went down to Nicaragua to assist them in the civil war. He and his men ended up defeating the opposition and instead of handing power over to Castellon, effectively took control of the country himself through a provisional president. US President Franklin Pierce recognized this government as legitimate.
Costa Rica, fearing that they would attempt to move southward, declared war on Walker and Los Filibusteros. Walker responded by organizing 4 companies for a total of 240 men to invade Costa Rica. Costa Rica then sent troops into Nicaraguan territory to meet them and defeated them at the Second Battle of Rivas.
One last fun tidbit: Juan Santamaria was a 25 year old drummer boy in the Costa Rican company who volunteered to approach the hotel where Los Filibusteros were staying and set it on fire. He succeeded in his attempt and drove the enemy out of their hotel, but was mortally wounded in the process. Santamaria is now recognized as one of Costa Rica's greatest heroes.
Edit: Dangit. After typing all of this, I looked at the question again and realized that OP wanted to know about small groups who conspired to start a major war NOT small groups who conducted wars themselves, like I thought on my first read of the question. Can I just leave this here as a "this isn't quite what you were asking but it's semi-related and kind of interesting"?
As a followup (if I may), wouldn't Gandhi & Jinnah (and Nehru) be held culpable for starting the Indo-Pak war(s) which specifically in 1971 could have easily escalated into World War III, what with a stand off between USSR & the allies (US and GB)?
A specific follow-up question (hope that's ok), would Henry Kissinger's actions to prolong the Vietnam war fit this description? All I've read on the topic seemed rather one-sided.