I think that the paper by Grigorchuk and Nekrashevych quoted by Kate does provide a requested example. Indeed, they construct (Sections 3 and 4) a non-amenable group which acts faithfully has a faithful self-similar action on a homogeneous rooted tree. This action is self-similar, so that it preserves the uniform measure on the boundary of the tree. Moreover, the orbit equivalence relation is a subrelation (mod 0) of the tail (or co-final in authors' terminology) equivalence relation. Since the latter one is hyperfinite ($\equiv$ amenable), the orbit equivalence relation is also amenable.
I think that the paper by Grigorchuk and Nekrashevych quoted by Kate does provide a requested example. Indeed, they construct (Sections 3 and 4) a non-amenable group which acts faithfully on a homogeneous rooted tree. This action is self-similar, so that it preserves the uniform measure on the boundary of the tree. Moreover, the orbit equivalence relation is a subrelation (mod 0) of the tail (or co-final in authors' terminology) equivalence relation. Since the latter one is hyperfinite ($\equiv$ amenable), the orbit equivalence relation is also amenable.