How to manage a bookings site after the owner leaves the organization

SubhamKSinha I tried all that but when I try to apply any changes, they don't apply.  I don't want to delete the site, I want to add someone else as admin.  The only way I've been able to get close to what I want to do, is to make a copy of the site, and then everything works fine.

SubhamKSinha 

Thanks. I missed that one :
Add-RecipientPermission -Identity <bookingmailbox@emailaddress> -Trustee <adminusers@emailaddress> -AccessRights SendAs -Confirm:$false

In summary :

1. Extract the list of all Bookings : get-mailbox -RecipientTypeDetails scheduling
2. Add this permission : Add-MailboxPermission -Identity <bookingmailbox@emailaddress> -User <adminusers@emailaddress> -AccessRights FullAccess -Deny:$false -AutoMapping:$false
3. Add this last permission : Add-RecipientPermission -Identity <bookingmailbox@emailaddress> -Trustee <adminusers@emailaddress> -AccessRights SendAs -Confirm:$false

And I create a little script to find booking without staff on it:

$AllBookingsMailbox = get-mailbox -RecipientTypeDetails scheduling

foreach ($BookingMailbox in $AllBookingsMailbox)
{
$Permissions = Get-MailboxPermission -Identity $BookingMailbox.Identity
$NbPermissions = $Permissions.Count
#If only 2 permissions that is the default ones
If ($NbPermissions -eq 2)
{
Get-MailboxPermission -Identity $BookingMailbox.Identity |Format-List
}
}

Kreera_House I struggled with initially to find the bookings mailbox name.  To find it, go to the Bookings Page and under "Share your page" you can see the email account name created for your site in the share URL.  The issue I had is these accounts don't show up in ECP, and initially I didn't realize they were a specific RecipientTypeDetails value.  

With that, you can also use this to get all your Bookings mailboxes:
Get-Mailbox -ResultSize Unlimited | Where {$_.RecipientTypeDetails -eq 'SchedulingMailbox'}