Renewal of Wedding Vows

Do you mean formally or just a nice public location?

I guess I could go for either one of those criteria. What place do you recommend?

How big will/should the crowd be?

There will be no crowd. I want to surprise my wife with this idea.

Would a famous and posh restaurant do the job?
https://www.sacher.com/en/hotel-wien/culinary-en-2/restaurant-anna-sacher/
When making the (required) reservation you can tell them the circumstances, and they may provide something special or a surprise.

Alternatively you may consider renting a car on Vienna's Riesenrad (giant ferris wheel):
http://www.riesenrad.co.at/vienna-ferries-wheel-rental.html

Or rent a tram:

http://www.rentabim.at/index.html

I just did this for a friend who is leaving Vienna for a new job. It was a hoot. You can bring your own snacks and booze and pick the route.

Or rent a tram:

http://www.rentabim.at/english/english.html
http://tram.at/?lang=en

You may choose freely the route, wherever tram tracks are. You need not to follow a route serviced by a regular tram line.

Great ideas everyone! Thanks so much!

If your wife likes gardens, and flowers are still in bloom when you're there, she might like the Volkgarten. They have a gorgeous rose garden. Another thought is at the Gloriette overlooking the Schönbrunn Palace.

Another thought is at the Gloriette overlooking the Schönbrunn Palace. Jean beat me to it. Or conversely, in the back of Schönbrunn looking up towards the Gloriette. Or if you want someplace a little more private, wander off into one of the little nook and crannies of the Schlossgarten.

I like the idea of a garden. Thanks!

wander off into one of the little nook and crannies of the Schlossgarten.

Especially into "The Maze". :-)

I like the idea of a garden.

If you really consider this, then go to https://www.wien.gv.at/stadtplan/en/ and enter "schloss schönbrunn" as search term, then zoom out at least two notches to have the whole palace garden on the screen. So you can see the layout. Clicking on "Luftbild" (top right) will give you an aerial picture (not from a satellite, but from a low flying airplane). You can zoom in then to see the details. Unfortunately the current picture was taken in winter, so nothing blooms.

An alternative is the Japanese Garden, often used to shoot after-wedding-pictures. Use "hohe warte 8" as search term, left to the red marker you find an area with a pond; that is the Japanese Garden. Zoom in to see the details.