The Lesser Holy Grail of Fuyuki (冬木の小聖杯?) is a magical construct designed to enable the descent of the Greater Holy Grail of Fuyuki in the Fuyuki Holy Grail War. The Lesser Grail is kept within the Holy Grail Vessel (聖杯の器?), an inanimate object or living being designed to intake the defeated Servants.
World of Fate/stay night
Up until the third Grail War, the vessel for the Grail was a literal "cup". However, during the events of the third War, it was damaged in battle and the ceremony was interrupted; for that reason, it had to be made as something that could manage and protect itself.
During the Fourth War, Irisviel von Einzbern was a homunculus forged by the Einzbern family to be the physical embodiment of the Lesser Grail. Her deteriorating condition as her consciousness was supplanted by the Grail was mitigated by implanting Avalon into her body until the final days of the war. When she was murdered by Kotomine Kirei, the Grail took on her persona.
During the fifth War, the Lesser Grail was made as the heart of Illyasviel von Einzbern, a special homunculus made with concentrated alchemy, with Irisviel as her mother and Kiritsugu Emiya as her father. The Grail hidden in Illya was even more refined than the one in Irisviel, proof that Illya was expected to be the Einzbern's trump card.
At the end of the Fourth War, Zouken Matou acquired the fragments of the destroyed Grail. He synthesized it with his Crest worms, which he put inside the body of Sakura Matou. Though she was incomplete, this turned her into a second Lesser Grail.
World of Fate/Apocrypha
 Fate/Zero material - Encyclopedia: Vessel of the Holy Grail [Other], p.098
Vessel of the Holy Grail [Other]
The substitute prepared to facilitate the descent of the Holy Grail. In the first three Grail Wars it was merely an object, but in the fourth war the Einzberns sought to gain an advantage in the struggle for possession of the vessel by implanting it with a will of its own and a sense of identification with the Einzbern family.
Although Irisviel's homunculus body is unmistakably organic, the inorganic Holy Grail was broken down and fused with her organs.
While it's certainly true that making the vessel required to win the war a member of their own faction is a reprehensible act of cheating, technically their only obligation is to "deliver a vessel to Fuyuki", which they did. Besides, it's a war. Is there really that big of a difference between stealing somebody's possessions and killing them then taking the body?
 Fate/complete material III: World material - Record of Heavens Feel - Holy Grail War: History of the Lesser Grail, p.009
History of the Lesser Grail
• Fourth Holy Grail War
• Fifth Holy Grail War
• Matou Sakura
In the final events of the Fourth Holy Grail War, Zolgen (Matou Zouken) acquired the fragments of the destroyed Grail. He combined the fragments with his Crest worms, which he put inside Sakura's body. Though she was incomplete, this turned her into a second Lesser Grail.