You have distribution groups setup in Active Directory that are synced with Office 365. In the O365 console, you cannot enable allow external senders as it is synced from AD. So you have to set this in AD.

There are a few ways to do this:

Active Directory

1. In Active Directory, make sure you have Advanced features enabled (Menu > View > Advanced Features).
2. Goto the OU that has the distribution group (do not search for the distribution group as you will not get the Attribute Tab when you run a search).
3. Goto the Attribute Editor Tab
4. Scroll down to msExchRequireAuthToSendTo and set the value to False

ADSIEdit

1. If you haven’t already connected to Active Directory open ADSIEdit and connect to Default Naming context
2. Navigate to the distribution group
3. Right Click distribution group and click properties
4. Scroll down to msExchRequireAuthToSendTo and set the value to False

Powershell

1. Run the following command
   
   PowerShell

   Set-ADGroup -Identity "SamAccountName of group" -Add @{"msExchRequireAuthToSendTo"=$true} -server "your AD server FQDN" -Credential (get-Credential)

   1	Set-ADGroup -Identity "SamAccountName of group" -Add @{"msExchRequireAuthToSendTo"=$true} -server "your AD server FQDN" -Credential (get-Credential)