Do I need an Austrian vignette?

Yes you do need one. Buy at the border station or at any filling station in Lochau (ca. 9E. for 10 days). Don't try to drive without, there are frequent checks any they have installed electronic controls at the entrance of the tunnel recently. The fine will be hefty (130+).

Yes, you will need a vignette (toll sticker). Buy the 10-day variant (€9.20).

If you are caught without it, either by a patrol or by a camera with license plate recognition, the fine is more than the price of a 1-year vignette.

Alternatively you can take route 190 to avoid the Autobahn (highway) in Austria.

In Switzerland you will need a vignette, too.

You only need a Vignette for driving on Autobahns, not normal roads.

If you take the Autobahn round Bregenz, you would need a Vignette. It is not "one long tunnel". There is a tunnel round Bregenz, however most of the route is in the open.
The crossing from Switzerland to Austria is at St. Margrethen and is not an Autobahn. You can buy a Vignette at the border post. You are not forced to buy one as that road is not an Autobahn. If you buy a 10-day Vignette it will also cover next Tuesday.

The alternative is to avoid the Autobahn and go through Bregenz, via the L202 and L190.

If your car was rented in Switzerland it will already have a Swiss Vignette, otherwise you need one of those as well (CHF 40).

Yes. We did this same trip last September. If you take the main route (which is the most likely), you do need the vignette. In my research, I read many "complaints from people who were ticketed on this small slice of road for not having it. Their complaints were 2 fold: "I was barely in Austria" and "there was no warnings that I needed it or that told me to buy it" . I researched trying to avoid the Autobahn, but it was slow and the couple of people I asked in Appenzell said just buy the vignette and save yourself a headache. There really are not very many warnings as you approach,(as Chris mentioned, we were not on an Autobahn as we approached) but I just pulled over at a gas station close to the border and got one. I also want to be legal and don't want the headache of getting pulled over and paying a fine when a little research will show me how to avoid it. Also, in my research, I noticed that many people who were pulled over in Austria, felt that the police "hover" at the border and look for people that do not have it. They called it a "scam", but I feel the only way you can be sure people follow the law is to enforce it.

Viamichelin will show you routes to avoid having a vignette when you click on the indicated box. They are, of course, slower than using the motorway, maybe adding an hour.

I saw a lot of road signs on the motorways in Switzerland, apparently for the benefit of Germans and Austrians, to the effect "Got your vignette yet?"

Thank you all so much for your help! I guess we got lucky the last time we drove through. We will need to get gas near the border anyway, so I'll plan to buy a vignette then. (We live in Switzerland and are driving our own car, so we already have the Swiss vignette.) Thanks again!

stopping to fill the tank in Austria will be a good idea as the Austrian fuel will be a lot cheaper than German and hugely cheaper than Swiss

The alternative is to avoid the Autobahn and go through Bregenz, via the L202 and L190.

It is still marked as B202 or B190 in most maps, including Google Maps.

A - Autobahn (highway, motorway) - blue square signs, bearing the designation Axx
B - (former) Federal routes - blue square signs, bearing the number, without the B
L - State routes - white oval signs, most routes do not have signs at all

To add another level of confusion:
In 2002 the federal routes were transferred by law to the states, which are forced now to pay for the maintenance. All Austrian states, except one, decided to keep the old numbering system and signage.

Only the state of Vorarlberg, with its capital Bregenz, had decided to convert all B routes to L routes, which in fact they are, legally spoken. A kind of protest against the unwanted gift they got from the federal govenrment.

As others said - avoid Autobahn. Vignette is mandatory on Autobahn. For that short distance in Austria it is not worth to buy it. I would rather detour on local roads.

As others said - avoid Autobahn. Vignette is mandatory on Autobahn. For that short distance in Austria it is not worth to buy it.

Hm... have you ever tried out that? The chances of getting in a real traffic jam on the route through the center of Bregenz are excellent. I know many people, including myself, who have done this only once and never again. Besides, the tunnel costs nothing, which is very unusual for Austria. Seen that way, €9 is a bargain. Will pay again on Friday. ;)

I've tried it and failed spectacularly. It took me an absolute age.