The origin of Mahayana is one of the genuinely open problems in the study of religion, and the honest summary is that the competing explanations are all still on the table.
The texts begin to appear in India around the turn of the Common Era. Beyond that, several accounts compete. One holds that the movement began among laypeople pressing against monastic control of the tradition; that case has weakened considerably, since the early Mahayana texts are monastic in tone and preoccupied with monastic concerns. Another traces it to the Mahasamghika, the party that separated from the Sthaviras in the earliest schism. A third, the forest hypothesis, places its origin among ascetics practising away from the great monasteries. A fourth points to the cult of the book: several Mahayana sutras command their own copying, recitation and veneration in terms that suggest a movement organised around particular texts rather than around particular institutions.
What is clear is that Mahayana did not begin as a separate community. For centuries, Mahayana and non-Mahayana monks are recorded living in the same monasteries under the same discipline, which is why the Chinese pilgrims who visited India could count them without being able to separate their institutions. The Mahayana was an orientation a monk adopted, not an order he joined.
That is also why the tradition has no founding date, no founder and no schism to record. It has an emergence, and the emergence is contested by people who agree about almost everything else.