@Jay: Either the Mail Recipients or the
User Options management role should work.
This command shows all role entries for the Mail Recipients management role. The output displays each cmdlet the role assignee can run, and also the cmdlet's parameters that the assignee can use.
(Get-ManagementRole "Mail Recipients").RoleEntries
The Set-MailboxAutoReplyConfiguration cmdlet is included in the role entries. You can also see the parameters of the cmdlet allowed for this role.
Or use this command to get a complete list of parameters allowed:
((Get-ManagementRole "Mail Recipients").RoleEntries | ? {$_.Name -eq "Set-MailboxAutoReplyConfiguration"}).parameters
Similarly, for the User Options management role:
((Get-ManagementRole "User Options").RoleEntries | ? {$_.Name -eq "Set-MailboxAutoReplyConfiguration"}).parameters