Are there any Nusach Ashkenaz Shuls that say Yotzros or Musaf on the weeks of the Four Parshios?
Yes, there are many who still do (though unfortunately a minority today). To name a few, Fifth Avenue Synagogue says the Musaf piyutim as does OZ on West Side (or at least they did last time I was there for Parshas Shkalim about 9 years ago), one or more of the Young Israels in the NY area do (I forgot which one but I know that at least one does it). In Brooklyn, there are a whole bunch of shuls that do so incl. nusach ashkenaz shuls (though the majority of those who say yotzros today daven sefard). Breure's in Washington Heights of course says all the piyutim as do a number of other shuls there. In Europe, many of the old shuls follow the minhagim as they were before the war and say yotzros (incl. the main shuls in London, Budapest, Strasbourg, Zurich, Basel...all of which daven ashkenaz)
Seems the Yekkis have
An interesting example is the Yerushalmi shuls in Israel, who follow the customs of Tamidei HaGra. (I think the Tukachinsky luach mentions this custom.)
They don't say Yotzrot or Krovot during the brachot of Shma or Shmone Esrei, but after Shacharit (and before taking out the Torah) they say the piyyutim.
I have heard that Ner Yisrael says the Mussaf for Shkalim and Hachodesh as do many shuls that follow the yeshiva's minhagim. Anyone have first-hand experience?
I also heard from a number of people (but again, I have not been there to confirm) that Lakewood says many of the piyutim for Arba Parshios. Anyone know exactly which ones they say?
I know that Chofetz Chaim Adas Bnei Israel in Baltimore says the piyutim in Shachris and Mussaf for Arba Parshios.
Kesher Israel in Washington, DC says just the Mussaf on Parshas Hachodesh (and not Shkalim), a remnant of the original custom at the shul.
All these places daven Ashkenaz.