A stacky Lie groupoid (in contrast to a Lie groupoid) is an internal groupoid in the category of differentiable stacks, such that the space of objects is an ordinary manifold.
Notice that differentiable stacks are equivalent to Lie groupoids modulo Morita equivalence.
Stacky Lie groupoids and their Lie theory were introduced and studied by Chenchang Zhu and collaborators.
Chenchang Zhu, n-Groupoids and Stacky Groupoids, 2008, International Mathematics Research Notices (2009) 2009:4087-4141; (http://arxiv.org/abs/0801.2057 arXiv:0801.2057). DOI: (http://dx.doi.org/10.1093/imrn/rnp080 10.1093/imrn/rnp080).
C.Z., Lie $n$-groupoids and stacky Lie groupoids (arXiv)
C. Z., Lie II theorem for Lie algebroids via stacky Lie groupoids (arXiv)
C. Z. Lie II theorem (pdf slides)
Henrique Bursztyn, C.Z., Morita equivalence of Poisson manifold via stack groupoids (arXiv)