How everyone joined the group
The history of the Rangers as a team is told in the five-part pilot To the Rescue. The two chipmunks Chip and Dale had a certain connection with crime-fighting from the very beginning. They were friends with the police dog Plato, and they accompanied him and Detective Donald Drake—who never learned that they exist—on some cases. But Drake's last case was to become the very first case of their own. He was falsely accused of having stolen the Klutchkoin Ruby, and with him and Plato locked away in a cell, the only ones who could solve this case were Chip and Dale whom Plato sent out to find Aldrin Klordane, the true thief who was officially pronounced dead.
So was Fat Cat, his feline companion who already got to know (and hate) this little team of amateur crime-fighters when they were only two. They met Monterey Jack and his friend Zipper at the docks shortly before Fat Cat, pulling off his own evil plot, sunk the chest that was Monty's home. So the main reason for Monty to join them was revenge on Fat Cat, and Zipper simply didn't want to go on alone.
When flying was required to continue the case, Monty led his new friends to an old bomber plane where he expected his old buddy Geegaw Hackwrench to live. However, Geegaw was lost more than a year ago, and the only one left was his daughter Gadget who first took over the pilot's job and then started a series of inventions which proved useful on the case. The Rangers were complete now when they encountered their second main villain, Norton Nimnul.
The name and the battle cry
As this case proceeded to its very final, the world's gold reserves suddenly lay in the hands of these five animals. And it was this moment when Plato gave the team the name Rescue Rangers, and just a bit later, Chip used the Rangers' famous battle cry, "Rescue Rangers, away!" for the first time.
A team forever
After they had successfully solved their first and biggest case and returned to the police station, the team was about to disband when they were sent onto their second case. A girl missed her dog and said that there was nothing but a hole in the ground where the doghouse used to be. This second case is not described any further, but it is likely that it made the bonds in the team strong enough to hold past its solving. Some time shortly after this case, the Rescue Rangers Headquarters may have been built into a tree in a park.
But the Rescue Rangers didn't stay a team of five all the time. The only member who has never quit is Chip, mainly because as the leader of the Rangers, he has to bear the responsibilities and just can't leave them alone, but also because more than any other Ranger, he has made a dream come true and a hobby of his a kind of a profession.
Dale quit in Le Purrfect Crime when he felt that he was nothing but a goofball for the others and temporarily joined Maltese de Sade. Chip could be said to have temporarily fired him, as he told Dale to go sightseeing.
Gadget quit in The Case of the Cola Cult when she lost her self-confidence as an inventor and was sure that her inventions were more of a danger than of any use for her friends.
Monty and Zipper already quit in To the Rescue (Part 4) after a dispute; Monty wasn't used to having a leader, and Zipper simply followed him.
Zipper quit once more in Zipper Come Home, thinking he was but a minor member of the Rangers and only useful for minor tasks, but Monty being about to sacrifice his own life for his friend's convinced him to come back.
Accepted Fan Fiction Conventions
Even in fan fiction, the Rescue Rangers hold together, and never (or rarely?) does a member quit forever unless he or she dies. The team rather grows; in some Rangerverses, the Rescue Rangers welcome new permanent members such as Foxglove, Tammy or Sparky.