The EXECUTE permission was denied on the object xp_availablemedia
The reason is that the user has enough permissions to perform a backup, but not enough permissions to enumerate the filesystem on the server, which is what SSMS does behind the scenes.

This behaviour is described in this support page, which suggests adding the login to the sysadmin role.

To work around it, add a backup device with sp_addumpdevice and let the user backup to that device.