Is all matter within the observable universe? The observable universe is everything within the range of the distance light can travel since the beginning of the universe. Since relativity states that matter cannot travel faster than light in any non inertial reference frame, and the Big Bang theory states that everything started out at a singularity, doesn’t this imply that matter could not have traveled further away from us than the border of the observable universe?
 A: 
The observable universe is everything within the range of the distance light can travel since the beginning of the universe.

No, it's the apparent distance of the most distant objects when the light that reaches us from them arrives here.  But something could exist which is further away ( = older ) from an earlier time closer to the beginning of the universe but the light from which either cannot be detected (is unobserved) or has not reached us yet.

Since special relativity states that matter cannot travel faster than light in any non inertial reference frame, and the Big Bang theory states that everything started out at a singularity

Before saying anything else I'd point you to this question Did The Big Bang happen at a point ?.  There are two particularly fine answers there well worth trying to get to grips with.
Special relativity is a useful theory for many purposes, but not for the Big Bang.  To explain (to some extent) the Big Bang and the expansion of the universe we need something developed from the Theory of General Relativity, which includes gravitation (something special relativity does not include).
The theory that describes the expansion of the universe is from a very clever idea called the FLRW metric.
This most definitely not special relativity and the key difference that explains your issue is that according to FLRW (when applied to our universe), spacetime itself expands as you get further away from a given point.  And to fit the data we see, it's not only expanding, it's expanding faster and faster.
One consequence of this is that while locally (in your almost flat space time close to you) the speed of light is fixed, this isn't the full story.  On cosmological scales FLRW means that space-time is also expanding and that this expansion can itself be faster than the speed of light.
One way to think of this is that in addition to their "local" velocity in their almost flat space-time close to them, objects are being "carried along" by the expansion of spacetime on a cosmological scale.

doesn’t this imply that matter could not have traveled further away from us than the border of the observable universe?

All of this expansion means that matter certainly could have traveled further away from us than the border of what we can observe.  It means a great deal more : as the universe expands the expansion will push objects away from each other faster and faster to the point where the objects they see eventually get too far away and are moving so fast that they won't be observable any more.
This Wikipedia page on the Cosmological Horizon explains these things in a more precise way.
