To create your Distrubution list with Send as Permission:
In your Control Panel
Click Exchange Hosting Service
Click Hosted Organizations
Click your domain name
Click Distribution Lists
Click Create Distrubution list
Click the "Mail Flow" tab uncheck "require that all senders are authenticated", save changes
Click the "Permissions" Tab and add needed user accounts.
Save changes

To receive email for this distribution list you will need to add the user as a member.
In your Control Panel
Click Exchange Hosting Service
Click Hosted Organizations
Click your domain name
Click Distribution Lists
Add Member
Save changes