How could it be possible to fully prevent wing tip vortex formation?
Wingtip vortices are an unavoidable side effect of producing lift with an airfoil of finite length (thus having "wingtips"). The wing is designed first to have relatively high pressure underneath the wing and lower pressure above it. Second, air passing over the top of the airfoil is directed downwards as it leaves the wing (thus "pulling" the wing upwards by Newton's Third Law). These two characteristics of an airfoil combine to make air around the wingtip want to "spin"; high-pressure air below the wing will "leak" upwards over the wingtip, then be directed back downwards along with the rest of the air, creating the wingtip vortex:



Reducing the pressure differential between air above and below the wing will by definition reduce lift, because it is this pressure differential producing lift. By the same token, reducing the "downwash" off the back of the wing will reduce lift because the force creating the downwash is also lifting the wing. Therefore, conventional wings will produce wingtip vortices by definition.

Theoretical ways to reduce this vortex include:


Make it harder for air to "leak" upward over the wingtip. This is the principle behind the "winglets" you see on many airliners; they prevent air iunderneath the wing being able to move over the top of it, thus reducing the severity of the rotation of the vortex (which reduces drag, which saves the airlines fuel):

 

Even small aircraft, like this PiperSport light sport, often have little blended winglets to reduce vortex drag, saving as much available power as possible from the limited-horsepower engines of this class:



The A-10's wingtip has a similar idea implemented the opposite way; a slight downward curve to the wingtip holds the "cushion" of higher-pressure air under the wing, reducing the amount of leakage around the wingtip, which lowers the wing's stall speed and also reduces tip vortices:


Don't have a wingtip. "Nonplanar" wing designs, including ring wings, reduce (but do not eliminate) vortex drag compared to conventional wing planforms simply by not providing a single border between lifting and nonlifting surfaces:



This is a concept drawing, but the idea of a tipless closed-loop wing has been incorporated into production small planes, like this Belarussian small single:



Vortex drag is not totally eliminated because there is still a downwash behind the wings relative to the undisturbed air to either side of the plane, but the intensity of the vortex (and thus the energy lost producing it which counts as drag) is reduced considerably. 
Reduce the pressure differential at the wingtip. This is a major way that swept, delta and double-delta wing planforms reduce tip vortices. Very simply, most modern wings on high-speed or high-performance jets have a smaller chord length and a thinner cross-section at the wingtip than the wing root. This provides less lift at the wingtip (and the wingtip stalls more easily and thus tends to stall first) than a straight wing with a relatively constant cross-section, but the reduction in pressure differential and downwash right at the wingtip reduces the two primary contributors to wingtip vortices: