Set an extension ladder against your house that is tall enough to extend at least 2 feet above the edge of the roof. Secure the bottom to the side of the house and the top to the roof fascia with ropes.
Climb on the roof and clear leaves, pine needles and other debris from around the vent opening. Reach into the opening as far as you can and pull out any debris lodged inside. If drainage improves after you do this, it may be all you need to do.
Screw a nozzle onto a garden hose and connect the hose to a faucet. Turn the water on all the way and climb onto the roof with the hose to clear obstructions deeper in the vent stack.
Point the nozzle into the vent opening and spray full force. Position someone in the house to watch the sink drains and toilets for overflowing while you do this. If any drain overflows, it means the obstruction is below it. The water will clear most obstructions, but stop spraying at the first sign of an overflow.
Use an auger to clear the vent if water overflows from a drain in the house or from the vent opening itself. Insert the auger head in the vent opening and push it in until you encounter resistance. Crank the handle to clear the blockage. You'll know it's clear, because the backed-up water will drain.