Lie groupoids serve as models for a certain kind of singular geometric spaces known as "smooth stacks". The classical orbifolds of Satake and Thurston provide examples of such spaces. Many more examples arise for instance in foliation theory, Poisson geometry or noncommutative geometry. Smooth stacks relate to Lie groupoids in much the same way smooth manifolds relate to differentiable atlases. Namely, there is a natural notion of equivalence between Lie groupoids, called "Morita equivalence", which has the property that two Lie groupoids are equivalent precisely when they model the same smooth stack. Motivated by the study of global geometric questions concerning the structure of differentiable stacks associated with proper Lie groupoids, we investigate the existence of multiplicative connections on such groupoids. We show that one can always deform a given connection which is only approximately multiplicative into a genuinely multiplicative connection. Our proof of this fact relies on a recursive averaging technique. We regard our results as an initial step towards the construction of an obstruction theory for multiplicative connections on proper Lie groupoids. Some applications of our results will be described.