Adding Send-As permissions to a contact in Office 365

First, make sure you have the contact in 365. Get the contact identity.

$UserCredential = Get-Credential
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection
Import-PSSession $Session
Get-Recipient

The identity is the name column.

Add-RecipientPermission "" -AccessRights SendAs -Trustee ""

User being the person that needs to SendAs.