How to use Azure Backup to backup files on on-premises Linux machine

You can use free VEEAM B&R https://www.veeam.com/linux-backup-free.html to automate the backup process itself and upload your backups directly to Azure VM mounted as a backup repository or automate/script the upload with azure cli.

Is it possible to backup individual files (i.e. database backups) from the disk of an on-premise Linux (CentOS) server to the Azure Recovery Services vault?

No, it is not possible. Based on my knowledge, you could backup Linux VM VHD with Azure Recovery Services vault, but backup a single file is not supported. Because if you want to back up files and folders to Azure, you need install and register the Recovery Services agent. However, the agent does not support Linux VM now.

Please refer to backup file or folder and how to restore file or folder on Azure.

Do I need the Recovery Services agent and is that available for Linux?

For now, backup agent does not support Linux OS.

Or is there a smarter way?

You could use azure cli to upload Linux File to azure storage account. Please refer to this link. Azure cli is supported on a Linux VM.