I have a shape file that doesn't have the elevation coordinates. I have the extracted x and y and with those I am looking for a way to solve for z or the elevation. Can this be done in FME, and if so with what transformer?

Try CoordinateExtractor to see if there are Z value store in the shapefile or not. If there's none, you will probably need to to do spatial filter/ relator with the known elevation file to see if it touch / equal / or intersect, you can merge the elevation from that file into your shapefile. Once that process is done, you can use 3Dforcer to apply that elevation into your shapefile (this will apply elevation to Z-value).

But short answer is, I do not know of a way that FME can apply elevation without elevation data being supplied.

The ElevationExtractor transformer is probably the easiest way to pull a z value out of your shapefile. That is of course if it exists, as @panda​  has mentioned.