What's the right way to clear a vent pipe?
One method used is to snake a garden hose up to the roof to where the vent line protrudes from  the roof. An expanding bladder of the size to fit inside of the vent pipe...



.. is attached to the hose and then pushed down into top of the vent pipe. A helper on the ground then turns on the water to the garden hose. The bladder expands to seal against the inside of the vent pipe and force the water to go down into the vent toward the plug up. 

Success of this scheme will depend to large degree if the blockage in the vent is a total closure of the pipe or not. A partial blockage may just let the water run by and not provide much help.