La Scala Tuxedo & Formal Gown Rental

I really don't think that is necessary. Only the season premier is super formal. June 20 will be end of the season. If it makes you feel more comfortable, wear a business suit and wifey in a nice black dress. Its not that big a deal. The most formal occasion I've attended in Europe is the Bayreuther Festspiel in August and even there, only 50% tuxes for the guys. I got by wearing a bow tie to try to blend in, but even that was probably unnecessary, as a significant minority wore jackets sans ties.

I see La Boheme is on the schedule for that night. Just read the paragraph regarding dress for the "House Rules", http://www.teatroallascala.org/en/box-office/prepare-your-visit/house-rules.html
You'll see that formal wear is not required. They just ask that you dress nice.

It is not necessary to rent a tuxedo ("smoking" in Italian) or any formal attire to attend the Opera.
Any shirt and a tie plus jacket and slacks will be more formal than many in the theater. Only at the Season "Prima" (first Season Premiere Opening) at La Scala, people dress very formally and the event has lots of VIPs in attendance. But you are not going to that event if you go in June (La Boheme is scheduled to be on that day).

I'll give you some websites below, but you can search more online buy googling these key words:
NOLEGGIO ABITI DA CERIMONIA MILANO
or
NOLEGGIO ABITI DA SERA MILANO.

http://www.centrosposiparadiso.it
http://www.lobosco.com
http://www.lariula.com/smoking.html

I am an opera worker so I understand your concern. A nice business suit and an elegant dress for your wife will be more than enough for La Scala. I would consider tuxedo only for the Dec. 7th premiere - and frankly speaking even for that a black business suit would be a very decent substitution. Personally, my tuxedo gets used only in Germany and Austria, that are more formal than Italy, black business suits managing the core of my work (occasionally tailcoat when playing on stage, but never required from the public).

As Roberto has pointed, a tuxedo is a tuxedo only on your side of the Atlantic pond. In Italy, France and Germany it is a smoking, in UK a dinner jacket.

Not only are a tuxedo and formal gown not required, they could also make you feel very self-conscious, as you would likely be the only people dressed in that manner.