These are the steps to take when you're not receiving the verification code via SMS:
- Make sure that you have a good cellular network connection on your phone.
- Check that your phone number is one that can receive SMS messages. You can do this by having someone send you a text message.
- Check that you have the correct phone number associated with your account:
- Devices using iOS 7.0.3 or later: Tap Settings > iCloud > Account, then tap Keychain. Check that the phone number listed under Verification Number is correct. If not, enter another phone number.
- Macs using OS X Mavericks v10.9 or later: Choose Apple () menu > System Preferences. Click iCloud, then click Account Details. Check that the phone number listed under Verification number is correct. If not, enter another phone number.
- Check Apple System Status to see if there is currently a service outage affecting iCloud Keychain.
- If you do not have access to a device that has iCloud Keychain enabled, you can contact Apple Support and verify your identity for help completing iCloud Keychain setup.
This solved my problem (after many minutes with Apple Support) but I don't know how to mark it that way. The authorization code link should really link where the contact Apple Support link goes rather than the generic support page.
By the way, neither I nor the guy on Apple Support could ever find a "Verification Number" anywhere on a device or on the "Manage my Apple ID" web site.