Getting a proxy account to backup to a share using Ola Hallengrens backup script
To make this happen, you need to give the computer domain account on which you have SQL Server installed rights to write to the share [Folder] on [ComputerName].

Say you have a computer SRV-SQLSERVER, and SQL services you run with NT Service*, but in some situation you can't create a domain account for them.
But you need to store backups on other domain server \SRV-BACKUP.

To have this working, create an share on \\SRV-BACKUP\Folder and give write permission for SQL box domain account DOMAIN\SRV-SQLSERVER$.

Some related info can be found here:

How to grant network access to LocalSystem account?