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.

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 direct sums of 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.

I don't know a precise reference for this folklore. The compact totally disconnected case is easily deduced from Pierce's Memoir of the AMS on modules over commutative regular rings. The modification for the locally compact case is known to those studying representations of reductive groups over local fields.

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 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.

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.