I can't speak from first hand knowledge about the Amex Air Miles cards (I have the Amex Gold Rewards and Platinum cards, i.e., the Amex Membership Rewards cards), but according to the referral section of Amex's site, you could refer your wife, and you would get 200 Air Miles for referring her. And yes, she would get the sign-up bonus once reaching the minimum spend. (Not sure if Amex Air Miles cards are similar to the Gold/Platinum cards, but with the Gold/Platinum cards, the sign-up bonus is actually higher if you use someone's referral link, rather than applying directly online.)
The American Express® AIR MILES®* Credit Card
As the Basic Cardmember, refer a friend and you1 could earn a bonus of 200 AIR MILES Reward Miles for each approved referral up to a maximum annual referral bonus of 3,000 AIR MILES Reward Miles. Offer subject to change at any time.