No, Dual Strike does not benefit from Vital Strike.
From the Dual Strike text (emphasis mine):
As a standard action, you can make two attacks, one with each weapon you’re wielding, applying the normal penalties for two-weapon fighting to each.
From the Vital Strike text (emphasis mine):
When you use the attack action, you can make one attack at your highest base attack bonus that deals additional damage. Roll the weapon’s damage dice for the attack twice...
From the PFSRD on "Attacks":
Most of the common actions characters take, aside from movement, fall into the realm of standard actions.
Making an attack is a standard action.
An attack action is a specific action described in the rules, and Vital Strike only triggers when you use this specific action. Dual Strike does not modify the standard Attack action; instead it grants you the ability to use a different action (presumably called Dual Strike). Because using Dual Strike is not using the Attack action, it does not benefit from (Improved) Vital Strike.
Unfortunately Pathfinder has a lot of feats whose design is... questionable at best, especially when it comes to prerequisites. As a result, you can't use the fact that feat A requires feat B as evidence that the benefits of feats A & B can be combined.
Why would you Dual Strike if it doesn't benefit from Vital Strike?
Dual Strike & Vital Strike multiply different parts of your damage. The Vital Strike chain multiples your weapon damage. Dual Strike, on the other hand, allows you to attack with both weapons when dual-wielding. If you're dual-wielding weapons with a small damage die (e.g. daggers, with a d4), then Improved Vital Strike would do 3d4+Str+other damage, while Dual Strike would do 2d4+(2*Str)+(2*other) damage. If your Strength bonus and other damage bonuses (flaming weapons, etc.) come out to more than the damage die of your main-hand weapon (and they probably do), then a Dual Strike would do more damage than an Improved Vital Strike.
I can't imagine a situation in which it would be worth spending all the feats to get Dual Strike, but if, inexplicably, you have Dual Strike, then using it is probably going to do more damage than using Improved Vital Strike. Unless you're doing something pretty weird, though, you want to full attack (since that does the most damage), and you're better served by investing your resources into getting Pounce (the ability to full attack on a charge) than by optimizing standard action attacks. Just like attacking twice (Dual Strike) did more damage than attacking once with some bonuses (Vital Strike), attacking many times (full attack) is better than attacking some smaller number of times with bonuses; it has the further advantage of not costing tons of feats.