If $A \to M$ is a Lie algebroid over a smooth manifold $M$ then a representation of $A$ is a vector bundle $E \to M$ with a flat $A$-connection $$ \nabla : \Gamma(E) \to \Gamma(E\otimes A^*). $$ If $G$ is a Lie groupoid over $M$ then a representation is a vector bundle $E\to M$ and a Lie groupoid homomorphism $G \to GL(E)$, where $GL(E)$ is the groupoid over $M$ whose arrows are linear isomorphisms of the fibers of $E$.

Now suppose $G$ is a Lie groupoid that integrates $A$.

Under what sort of conditions can every representation of $A$ be integrated to a representation of $G$?

Two extreme cases are well-known:

If $M$ is a point then we're just talking about Lie groups and algebras and every representation of the algebra integrates to a representation of the group if the group is simply connected.

If $A = TM$ is the tangent Lie algebroid and $G$ is the fundamental groupoid then the Riemann-Hilbert correspondence says that every representation of $TM$ (i.e. flat vector bundle over $M$) corresponds to a representation of $G$. On the other hand, the pair groupoid also integrates $A$ but it doesn't seem like it has the same representations (I think a representation of the pair groupoid is given by a vector bundle $E\to M$ and a section of $p_1^* E^* \otimes p_2^* E \to M\times M$, where $p_j : M\times M \to M$ is the projection on the $j$th factor).

I would be interested in seeing any references that talk about this.