Actually you can with SteamPlay. Though you won't be able to swap mods under Linux, whatever you've installed with NMM will work, even SKSE. You just need to rename skse_loader.exe to SkyrimLauncher.exe. Also, SSE doesn't work yet. So, there's that.
So is this going to be a NMM clone or a MO clone? Linux has in-built support for OverlayFS, which is much better than MO's USVFS for windows. So I really hope they take that into consideration when making this.