As a complement to the answers above : it is kind of well-known (at least I thought it was) that the natural morphism
$$\operatorname{\mathbf{Hom}}_{gr} (G,H) \to \operatorname{\mathbf{Hom}}(BG,BH)$$
is an $H$-torsor (where $H$ operates on $\operatorname{\mathbf{Hom}}_{gr} (G,H)$ by conjugacy). In other words, the internal Hom stack $\operatorname{\mathbf{Hom}}(BG,BH)$ identifies with the quotient stack
$$\left[\operatorname{\mathbf{Hom}}_{gr} (G,H) \right/H ] \simeq \operatorname{\mathbf{Hom}}(BG,BH)$$
of the Hom sheaf $\operatorname{\mathbf{Hom}}_{gr} (G,H)$ by the conjugacy action of $H$.
This of course answers the original question completely (globally, no; locally, yes).
The proof is straightforward enough. For instance here is a sketch off the local surjectivity: let $\alpha : BG\to BH$. If $s_G$ is the canonical section of $BG$, we get a morphism $G=\operatorname{\mathbf{Aut}}(s_G)\to \operatorname{\mathbf{Aut}}(\alpha(s_G))$. Since $\alpha(s_G)$ is isomorphic to $s_H$ locally, this defines locally a morphism $G\to H$, up to conjugacy by $H=\operatorname{\mathbf{Aut}}(s_H)$ of course.
I have learnt this from Angelo, and since this is nice and quite useful, I am happy to share it back with you. For the context: this isomorphism is a frequent guest in non-abelian cohomology, and in tannaka duality.