I also forgot my residency permit once. But I had a photocopy.
The border officers have the capability to look it up at the border, but they don't like to do this because it takes longer and stops the flow of people in the already impossibly long line.
They will reluctantly roll their eyes and lecture your wife on how you always have to carry it with you because it is "super important" plastic.
But, if they can take the numbers/info from the photocopy, or the email or whatever your wife can produce at the border, and go back into the back and look up the visa and your wife will *probably* be allowed to carry on her way.
This makes me wonder why we even have to carry them at all. Surely all our travel documents could be pulled up from a database when we scan our fingerprints or face or both... I give it ten years