This answer is not for vertex transitive hypergraphs (I have not noticed that condition)!

No simple necessary and sufficient condition can exists as 3DM is NP-complete:
http://en.wikipedia.org/wiki/3-dimensional_matching

Of course, if you are only looking for a sufficient condition, one can come up with several, eg. see: http://arxiv.org/abs/1101.5830 where it is proved by Imdadullah Khan that "A perfect matching in a 3-uniform hypergraph on $n=3k$ vertices is a subset of $\frac{n}{3}$ disjoint edges. We prove that if $H$ is a 3-uniform hypergraph on $n=3k$ vertices such that every vertex belongs to at least ${n-1\choose 2} - {2n/3\choose 2}+1$ edges then $H$ contains a perfect matching. We give a construction to show that this result is best possible."

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No simple necessary and sufficient condition can exists as 3DM is NP-complete:
http://en.wikipedia.org/wiki/3-dimensional_matching

Of course, if you are only looking for a sufficient condition, one can come up with several, eg. see: http://arxiv.org/abs/1101.5830 where it is proved by Imdadullah Khan that "A perfect matching in a 3-uniform hypergraph on $n=3k$ vertices is a subset of $\frac{n}{3}$ disjoint edges. We prove that if $H$ is a 3-uniform hypergraph on $n=3k$ vertices such that every vertex belongs to at least ${n-1\choose 2} - {2n/3\choose 2}+1$ edges then $H$ contains a perfect matching. We give a construction to show that this result is best possible."