One expects a notion of volume measure on Lie groupoids (also known as differentiable stacks), which generalizes the notion of groupoid cardinality of finite groupoids in that it reduces the volume of the object space by the degree to which automorphisms encode weak quotients.
A solution to this was proposed by Alan Weinstein, and re-interpreted in terms of 2-vector bundles by Richard Hepworth.
One would expect some relation of this to the Lagrangian BV formalism, which is also a formalism for integration over $L_\infty$-algebroids.
Alan Weinstein, The volume of a differentiable stack (arXiv)
Richard Hepworth, 2-Vector Bundles and the Volume of a Differentiable Stack (pdf)