Hello :)
I had to google when exactly that day was (June 7th, right?).
Stores will definitely be closed, but I am assuming
Vegetasia (all you can eat taiwanese buffet),
http://tupalo.com/de/wien/vegetasia-ungargasseand it's cousin restaurant
Xu's Cooking:
http://tupalo.com/de/wien/xus-cookingwill be open. They are always open!
Yamm
http://www.yamm.at/ is open on holidays from 10 am to 6 pm. It's on the pricey side (you pay per weight) but vegan options are clearly marked on the buffet.
Museumsquartier
http://www.mqw.at/ will be open, here are prices and opening hours:
http://www.mqw.at/en/visitor+service/ho ... titutions/ (I like the MUMOK and Kunsthalle best but if you are into Schiele and Klimt, go visit the Leopoldmuseum).
Albertina will be open too:
http://www.albertina.at/I am assuming there will be some sort of big Catholic spectacle going down by St. Stephen's (Stephansdom) but Austria is a Catholic country, so most churches will have a procession.
Let me know if you have any other questions.