Roy,
Obviously you don't want to kill the birds in their nest - that creates an even worse problem. Likewise you don't want to drive off the parents until the baby birds can also leave the nest. People make these mistakes with squirrels in the attic and then they wind up with dead & decomposing adult squirrels and/or babies in the attic.
I'd wait until the family of birds are all old enough to exit on their own and then I'd drive them off by venting something slightly noxious through the vent. A very slight wiff of tear gas or pepper spray might work. Even steam from boiling water on the stove might make it uncomfortable enough to make them leave. The possibilities are endless, but just don't kill them in the ductwork.
I'd replace the vent door with one which prevents entry. Or remove the old one and rig a screen on the inside of the vent assembly. In either case, I'd fish out the nest while the vent door is removed. I'd follow that with a very long blast of compressed air from the range hood side to attempt to blow out as much debris as possible.
Good luck, Gideon