Can an Alchemist use a launching crossbow with their bombs?
Not by RAW
The Alchemist' Bombs ability states:
Bombs are unstable, and if not used in the round they are created, they degrade and become inert
So the alchemist would create its bomb, load it, and by the time he fires it it has become inert.
With the Delayed Bomb discovery:
The alchemist can place a bomb so that it explodes a number of rounds after the alchemist ceases contact with the bomb.
It is not clear if it is possible to "place" it as an ammunition of the crossbow.