Can an object move without relying upon Newtons third law? Pretty much everything we do appears to boil down to practice of Newton's Third law. 
Even this rug I'm seated upon relies upon reaction from the floor to keep me seated. Amongst others the same reaction helps me move, sleep/recline .. and so on.
What I wish to know - is it possible for a body to move without relying upon a reaction from the objects/whatever in the direction it wishes to move?
 A: The principle behind the third law is the conservation of momentum (see also the last paragraph in this section on the third law).
If you start with your stationary mass its momentum is zero. If it could start moving on its own the momentum would now be non-zero i.e. momentum wouldn't be conserved. Noether's theorem tells us that conservation of momentum is the same as saying that the laws of physics are unchanged by a displacement in space, so it's a pretty fundamental aspect of the universe. In this case it means your mass can only start moving if something else moves in a different direction to keep the total momentum at zero.
So the answer to your question is no!
A: John Rennie answer is almost correct, but there is a small (thirty orders of magnitude-small to be precise) counterintuitive exception to this rule when the spacetime is curved over the dimensions of an extended body. It is called spacetime swimming, which is basically a way to get a net displacement by doing internal displacements in a noncommutative order.
However the effect is extremely tiny, even in strong gravitational fields, net displacements that can be achieved in a shape cycle are of the order of $10^{-30}$ meters
Here is a reference to the original paper: http://dspace.mit.edu/handle/1721.1/6706
