Please check VMware documentation to confirm if the Storage/Backup API is covered by the Essentials license (as this is one of the requirements for Backup Exec to be able to do backups. Also check the Backup Exec SCL for the BE version you are using to make sure it support your ESXi version.
VCenter is not a requirement (although if you have a VCenter Managing your ESXi hosts you should configure the backups via the Venter and not directly) Backup Exec can be configured to backup and restore directly from unmanaged/stand-alone ESXi hosts
Finally vSphere and ESXi kind of refer to the same thing (just different terminology) so vSphere is present whether or not vCenter is present so your vSphere query does not apply.
Thank you for the prompt response, it is appreciated.
Can you provide a link for the documentation around:
"Backup Exec can be configured to backup and restore directly from unmanaged/stand-alone ESXi hosts"?
I don't know that we have a document but you basically configure in the same way as a VCenter except when you first add the ESXi environment into the BE console, you specify the (resolvable) DNS name or IP address of the ESXi host and you use the root account of that host to login.