Hotels with an Elevator(Lift) in Florence and Venice

No, many, do have elevators especially in the price range you mentioned, which is at least 3 star if not 4 star hotel.
However I don't know the extent of your wife's mobility. A bigger issue might be the extent of walking that Venice requires, especially considering that all numerous bridges you need to cross have a few stairs to climb. Getting on and off boats may also present a challenge depending on the mobility impairment.

In addition to hotels, you can find many 'lift served' apartments in that price range. If there are mobility limitations, the extra space in an apartment may be very welcome. Only in Rome did we pay more than $200 a night for a 2 bedroom/2 bath 'lift served' apartment with A/C (and we splurged in Rome because of the apartment and its location). Really nice 1 Bed/1 Bath apartments can be had in each city center for under the low-end of your budget.

I took my mother, who also has mobility limitations, to Venice-Florence-Rome last fall and Venice was a real challenge. The vaporetto stops in front of the Santa Lucia train station and at Piazza San Marco can be incredibly busy. I would plan your stay to be in close proximity (and on the same side of the canal) to a vaporetto stop serving the routes you are most interested in using. Google Maps does a pretty good job of helping you in this situation.

Even if the building has an elevator, there may be steps to reach it, either outside the building or inside. When you narrow your list, you should double-check with the property about how many, if any, stairs there will be.

To start, many places are listed in booking.com or other sites; toward the bottom of the listing/reservations area, look at the "facilities" section; if they have an elevator, it will be listed.

Keep in mind that many restaurants have bathrooms up or down a flight of stairs.

Moving about Venice should be an bigger concern than just fonding a hotel with elevator. You need to find a hotel near a vaporetto stop so that she doesn't have to climb the stairs up and down multiple bridges over the canals. Trip Advisor has an article on Venice for the Disabled which will be belpful to read.
I remember hearing about a hotel steps from a vaporetto stop. There are likely many of these but they probably not budget hotels either. Good luck in finding a solution. It is a magical place to visit.

The Hotel California has an elevator. We stayed there on our RS Best of Italy tour. It has a great location between the Duomo and the Accademia.

I'll second the Hotel California - it is the perfect location IMHO.
They have a wonderful terrace on the roof where you can sit and enjoy the evenings with a nice glass of wine.....we did almost every evening before dinner.

Suki, you made a great point about the need to find a hotel near a vaporetto stop. I am hoping someone will give some names.. I do not mind exceeding my budget for this one.

Thanks

Locanda Ca' San Marcuolo is about 50 meters from the San Marcuolo vaporetto stop, it has an elevator (big enough for two). In 2014, I paid less than €150 for two.

Florence - hotel Pendini has an elevator. Excellent location for Florence. Just request a quiet room in back. Lovely breakfasts. I stayed here twice and would gladly stay here again.

In Florence I also recommend the hotel California. Such a lovely place. It has mirrors on the ceiling and they serve pink champagne on ice.
if you have an early flight or train give yourself ample time, because at the hotel you can check out any time you want but you can never leave.

Hotel papadopoli in Venice. It's near Piazzale Roma, so you don't have to lug your bags too far.
http://www.accorhotels.com/gb/hotel-1313-hotel-papadopoli-venezia-mgallery-by-sofitel/index.shtml

Check out http://europeforvisitors.com/venice/bridges/. It might help you find a hotel which suits your needs.