Cap an empty vial, open the oven, and stick it in the position under the probe (3 oclock). Look on the back of the HS for the vent (a metal tube sticking out next to where your gases are plumbed), you need to connect a piece of tubing to that because to test the valves you stick the vent into some water and we're going to check for bubbles. The headspace comes with a piece of tubing but you may have lost it, if you have an LC the waste tubing is a similar size.
Once you have some tubing press the advanced function button, then 5, then enter. You'll use the arrow keys to go back and forth, and the 1 and 0 buttons to change settings (1 up, 0 down for the lifters and 1 on, 0 off for valves).
First press the arrow until you see the sample lifter, press 1 in order to push the empty sealed vial up onto the probe. Next press the arrow until you see the vent valve, the vial pressure valve, and the sample valve.
The first thing you do is pressurize the vial (vent closed, vial pressure open, sample closed) and make sure you don't see bubbles. If you do your vent valve is stuck open.
Once the vial has pressurized for 15 seconds or so, turn vial pressure off and turn vent on. You should see a nice stream of bubbles which comes to a stop in 5-10 seconds. If it continually bubbles your vial pressure is stuck open, if you get no bubbles your vial pressure is probably stuck closed and you're only getting tiny peaks because when the vial heats in the oven a bit of pressure builds and a small amount of sample makes it into the sample loop. I suppose its also possible that no bubbles could indicate a vent valve that is stuck closed--if you have vent and vial P on at the same time you should see a constant stream of bubbles which would prove you're getting vial pressure. I usually just replace both valves if the test fails.
If everything looks good go ahead and repeat the process with the sample valve open.
Are you using an EPC or manual pressure control?
One time I saw a headspace where the vial pressure had recently been replaced but they were still seeing very small peaks, and it passed this test. I took the probe out and stuck it through a sealed vial half way filled with alcohol and inverted, solvent dripped through the needle just fine--proving that the probe wasn't plugged. Next I examined the zero dead volume union that is on top of the probe, and it was fine. Then I took an autosampler needle and unscrewed the sample loop, I injected 5 uL of alcohol from the needle directly into the loop and then screwed it back on. When I turned the pressure back on I got a nice gigantic peak after about 5-10 seconds, indicating that there were no plugs from the loop all the way through to the detector (and it was a huge peak, and system already passed leak test so no leaks!)
This indicated to me that the fault was the piece of tubing going from the probe to the 6 port, I replaced it and the problem was gone. My guess is that when the vial pressure failed they did a bunch of troubleshooting, trying to inject samples. Each sample would heat up, and 'burp' a small amount of sample when the loop was opened in order to sample. Some of the sample somehow solidified in the tubing, despite it being close to the various heated zones.
edit: and don't forget to take the empty vial out of the oven! Otherwise once you get things fixed you'll go to run some samples and the first 3 or 4 vials will go in, then it will try to put one on top of the empty vial and screw up your sequence

.