Can we use BLS as a VRF?
Yes, in the random oracle model, the hash of a BLS signature makes a VRF essentially as secure as the BLS signature scheme (provided the verifier accepts only the unique canonical encoding of each signature).

This works because BLS signatures are unique.  Fix a pairing $e\colon G_1 \times G_2 \to G_T$ on groups $G_1$ and $G_2$ of prime order.  For any fixed $A \in G_2$, the homomorphism $\phi_A\colon \sigma \to e(\sigma, A)$ is an injection (or the system is trivially insecure) because $G_1$ has prime order.  Let $B \in G_2$ be the standard base point.  A putative signature $\sigma \in G_1$ under public key $P \in G_2$ on a message $m$ satisfies the signature equation $$e(H(m),P) = e(\sigma,B) = \phi_B(\sigma).$$  Since $\phi_B$ is an injection, there is at most one possible signature $\sigma$ for any public key $P$ and any message $m$.