To get QuickBooks to send email via MAPI to the Click-To-Run version of Outlook (2013, 2016, 365), we consult the amazing Jared Dexter, who found the following.

1) In the Win.ini file add the following section:

[MAIL]
MAPI=1
MAPIX=1

2) Open the Registry Editor and add these keys and values. Some or all most likely won’t already be there:

[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\MAIL]
MAPI=1
MAPIX=1

[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows Messaging Subsystem]
MAPI=1
MAPIX=1

[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows NT\CurrentVersion\MAIL]
MAPI=1
MAPIX=1

[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Microsoft\Windows Messaging Subsystem]
MAPI=1
MAPIX=1

Open QuickBooks, Outlook should now be an option. This has been seen to work in Windows 7 32-bit and 64-bit, Office 2013 Home & Business 32-bit and Office 2016 via 365, with QuickBooks Enterprise 14 and 15.