The standard Arrows
Oliver machines his own arrows, made out of typical metal materials. There's nothing to really trace back to.
Oliver's primary bow early on is the one he returned from Lian Yu with. There's nothing to trace back to that could connect to him.
He does receive another bow later, but from sources not connected to him directly.
The gadgets and gadget-based arrows
During the first season, Oliver does use some fancy technology and arrows that he likely did not create himself (the hacking arrow, for example). Later on, after Felicity joins the team as a full-time member, she handles most of the technology, but the early tech is largely unexplained.
However, as we learn in later seasons,
Oliver has worked for/with ARGUS in the past.
It is possible that some of his early technology and gadget arrows are leftovers from those days. Given the nature of this source, much of those items will be largely untraceable and unable to connect to Oliver.