I am wondering whether there is a Lie algebraic version of principal bundle for Lie group over a given manifold $M$. The first thing I try to think of is group cocycle picture of principal bundle.
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A very comprehensive source is: MR2157566 Mackenzie, Kirill C. H. General theory of Lie groupoids and Lie algebroids. London Mathematical Society Lecture Note Series, 213. Cambridge University Press, Cambridge, 2005. xxxviii+501 pp. Starting from the total space, there is also the notion of the action of a Lie algebra on a manifold, which one can extend to a Lie group action by enlarging the manifold (and loosing Hausdorff in general), see here. 


Building upon Peter's answer, An Atiyah algebroid, or transitive Lie algebroid is one answer to your question. 

