Google does not give a definitive answer as to how I can verify a business in GMB without postcards. It says that some businesses MAY be eligible for phone/email verification. Also, some businesses MAY be eligible for instant GMB verification through Search Console. But there's no definitive answer.
I remember reading somewhere that this workaround can be useful:
Create listings on business directories such as YP, CitySearch and Yelp.
Google will then automatically create a Maps listing for your business using that information, within 24-48 hrs.
Verify that Maps listing with your website's Search Console email address. This way, you (probably) won't need postcard verification.
How would you go about verifying a business in GMB without postcards?