I just saw this question now.
If $X$ is a Hausdorff space with a basis of compact open sets, then the category of sheaves of abelian groups on $X$ has enough projectives. Let $R$ be the ring of locally constant functions $f\colon X\to \mathbb Z$ with compact support and pointwise operations. $R$ is a unital ring iff $X$ is compact, but it is always a ring with local units (i.e., a directed union of unital subrings). An $R$-module $M$ is unitary if $RM=M$. This is equivalent to for all $m\in M$, there is an idempotent $e\in R$ with $em=m$. It follows that the category of unitary $R$-modules has enough projectives (the modules of the form $Re$ with $R$ idempotent do the job).
It is known that the category of sheaves of abelian groups on $X$ is equivalent to the category of unitary $R$-modules. The equivalence takes a sheaf to its global sections with compact support. Thus the category of sheaves of abelian groups on $X$ has enough projectives.