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I have a set of points that are the vertices of a polygon. Those points are in the same (arbitrary) plane. I want to move/rotate them so that the polygon sits in the $xz$ plane, (so that all vertices have a $y=0$ coordinate) without deforming it.

I guess the method would imply finding the plane coordinates, finding out which transform puts said plane in the $xz$ plane, and then using that transformation for all points, but I have no idea about how to do it in practice.

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    As you can shift/rotate the resulting polygon in the $xz$-plane, there are multiple solutions. – 2017-01-10

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You can find a vector perpenduicular to plane by calculating the cross product of two vectors between your points, then you just need to find a transformation that transforms that vector to something parallel to $(0,1,0)$, that's a well known operation.