From Italy to Austria during winter, by car

In Austria (and in Germany!), it's actually not mandatory to have winter tires or any form of special equipment if the road is in good conditions. The rule only kicks in in “winter conditions” (“winterliche Fahrbahnverhältnisse”), i.e. if there is actually snow or ice on the pavement. Except if you are driving in the middle of the night on a day with heavy snow fall, it should not be the case on the main motorways, even during the winter.

As far as Austrian law is concerned, “winter tires” are not necessarily snow tires (with the “alpine” sign) but tires with the “M+S” sign, which is also the case for many “all-season tires”. You could check with rental companies if their cars have those.

Alternatively, it's also possible to use “snow chains” instead of special tires but that's only allowed if the road is completely covered in snow.

You do have to pay for the yearly vignette for Switzerland (Austria also has a vignette but sells short-term ones).

Are you sure that the Italian car rental agency does not offer cars with winter tires? I really doubt that, because also in Italy it gets very cold in the winter and there will be snow on the roads. Maybe the cars have all year tires? Have you already talked to the car rental agency?

Considering the vignette: In Switzerland you pay 40 Swiss francs for 1 year and you can't avoid that if you want to drive on the high ways. You could probably risk to drive without a vignette because checks are rare, but if you get caught, the fine will be quite high. Actually the fine is 200 Swiss francs, and you're required to buy a vignette for 40 Swiss francs on the spot. If you're lucky, the car already has a vignette from 2014. With a 2014 vignette you can drive on Swiss highways until 31. January 2015.

Concerning Austria, you also have to pay for using the highways. The shortest amount for the road toll sticker is 10 days and it costs 8.50 Euros. Be aware that some parts of the highways still cost extra money, e.g. the Brenner Autobahn (the highway you're taking when going from Italy to Innsbruck), Hohe Tauern Autobahn, Arlberg Tunnel, Karawankentunnel, and so on.

An important note about the Swiss Vignette: It is not legal to slightly stick the vignette to a car, remove it, and use it on another car.

That said, unless you want to make a gift to the rental car company, make sure that they provide a car with Vignette, or that they reimburse you if you have to buy one.

To avoid the Vignette, just don't drive on the highways (green signs, A- numbers). However, this will reduce your options to travel through Switzerland to Italy, because the two common routes (Gotthard and San Bernardino) are highway, and the passes are closed.

The most likely open route would be (Landquart)-Klosters - Vereina tunnel car train - Engadin valley - Maloja pass - Chiavenna - (and further on), or … Engadin valley - Bernina pass - Tirano - (and further on).

Also possible is Martigny - Grand St. Bernhard - Aosta

And finally …-Spiez - Frutigen - Lötschberg tunnel car train - Goppenstein - Brig - Simplon pass (if open) - Domodossola - (and further on). If the Simplon pass is closed, there are some Simplon tunnel car trains

In any case, stay informed which routes are open; it may change from day to day, maybe even from hour to hour.

And a final comment about winter tires: At temperatures below 7 degrees centigrade, they provide better traction than summer tires. And if you have an accident with summer tires when winter tires would be needed, the insurance may limit the coverage.

Can you check with rental companies if they offer Snow Chains as an option? If not, you may buy them yourself (prices start from 50-60 euros).
According to austria.info "Mandatory Winter Equipment" , you should be ok since winter tires or tire chains are required only if winter conditions arise. You can mount them at occurrence.

Be aware that a "vignette" is required in Austria too, but is much less expensive than italian "pedaggio" on the "Autostrada".