I can do a kong over something about the size of a picnic table, however I notice that on anything longer, my legs start to fall, and I have to bring my feet down.
However many traceurs can get themselves much higher than an average person, and yet are not even particularly strong.
What technique do traceurs use to get themselves that high?
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Also look into going to a gymnastics center for some basic classes. There are a lot of gymnastic techniques that are essential for a good traceur, such as rolls, body orientation, proprioceptive/kinesthetic awareness, weight transition, etc.
One of the things that is going to help you in your kong along with the center of gravity noted in the other answer is a concept called "blocking". This is being able to take momentum in one direction and redirect it in a different direction.
This is best exemplified by the vault in gymnastics. The gymnast sprints down the runway, gathering horizontal momentum. They hit the springboard, which starts the transition to vertical, and then their arms plant and "block" the momentum into a vertical direction. This is going to be a vital skill for any traceur.
By dropping your chest lower than your butt, your center of gravity stays lower. This makes it easier.
Overall you're not jumping higher, just your legs are. When your hands hit, you have to push the ground away to get your chest high enough again to not face-plant.
As a fellow traceur, this is one of the vaults I've had most difficulty in achieving. The type of kong vault you are referring to is commonly known as a diving kong. To perform a diving kong it's all a matter of progression, technique and commitment.
In order to properly perform a good diving kong, you must take off from a split-step instead of a punch (a punch take-off is primarily used on high objects, where a split-step is used for diving over long objects), throw your arms forward and lift your hips above your head. Only at the end of the vault should you then plant your hands to follow through and land.
Practice using the diving technique on small objects first, by taking off from further away, then slowly make your way up to larger vaults until your able to do a full picnic table length.
While most of this has been covered in one form or another, I'd also recommend Parkour Science's video on the Kong vaults. The one thing that I think isn't being entirely covered in the prior answers here is that it really is a parabolic motion. In the Dive Kong, you land your hands as you're descending from the peak of your jump, which naturally keeps the legs higher.
