Ambition in this novel takes the form of dreaming. George and Lennie hold to a dream of "getting a stake" by working at the ranch (or other ranches) so that they can buy a small ranch of their own.
This dream is eventually shared by Candy and Crooks as they also enjoy the idea of 1) having a piece of land of their own and 2) having a life defined by friendships, permanence and peace. These qualities of life are seen to be lacking when we meet these characters in the novel.
The ambition represented in this dream of land ownership is often aligned with the American Dream and in the context of the novel represents a complete departure from the limits and deficiencies of the lives the characters currently lead.
George and Lennie dream of owning a farm, but by the end of the novel the dream has failed. Their plan is doomed because human fellowship cannot survive in their world and also because their image of the farm is overly idealized.
In terms of ambition, we can see this dream as an example of a desire for social and financial "upward mobility" as the characters imagine working for themselves and not for a boss.
Similar dreaming is found in Curley's wife. Her ambition was to get to Hollywood and become a star. She has more-or-less given up on this ambition as a possible reality but still clings to it as a cherished vision.