would the laser light line continue to exist between the 2 mirrors if the source of the laser stopped.
As DJohnM commented,
light travel to the moon and back, even though the laser source is off for most of the round trip.
This happens because any light source emits photons. They are indivisible packages of energy and they are traveling as long as they not get absorbed by an obstacle or - more precise - by a subatomic particle.
Would it just continue existing between the mirrors using itself as the source or is this not possible for light?
As told above, once emitted, the photon is on it own and don’t anymore care about the source.
But where is another point about the mirrors. What garyp told you in his answer is about the technical imperfection. Beside this, any reflection process of photons is accompanied at least with the transfer of a momentum to the mirror. Every photon pushes the mirror a little bit and any iphotonphoton - due to energy conservation - leaves the mirror with a lower frequency. So the light dies a infrared death and the mirror gains velocity, or the temperature raises.