Is it possible to sort droplets inside an oil phase with a FACS
It's possible that there's some specialized modified machine that can do it, but conventional flow cytometry machines can't do it. This article suggests one possible approach:


  Unfortunately, FACS instruments are incompatible with nonaqueous suspensions, so to sort a water-in-oil emulsion, it is necessary to carry out a further emulsification to produce a water-in-oil-in-water double emulsion. The resulting sample, now dispersed in an aqueous phase, is amenable to FACS sorting.


--One in a Million: Flow Cytometric Sorting of Single Cell-Lysate Assays in Monodisperse Picolitre Double Emulsion Droplets for Directed Evolution

There are also microfluidics machines that might be able to do it, but I think you'd have to build your own rather than being able to buy a solution (which would be expensive in any case, I think).