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This is a question regarding Joomla! ACLs, not really something about Akeeba Backup. You have to make sure that the users belong to groups which give them both Access Administration Interface and Backup Now privileges for Akeeba Backup. Login as a Super Administrator to your site and go to Components, Akeeba Backup. Click on Component Parameters. Click on the Permissions tab. Click on the group you want to manage. The five permissions are:
- Configure (the first one): this allows users to access the Component Parameters
- Access Administration Interface: self explanatory; if it's not allowed, you cannot access the component
- Backup Now: allows users to run backups
- Configure (the second one): allows users to configure backup profiles and have access to pretty much all other buttons on Akeeba Backup's main page.
- Download: allows users to download backup archives
At a minimum you need Access Administration Interface and Backup Now to take backups. You may also need the Download privilege if the user is supposed to be able to download the backup archives through the browser (which is always a Very Bad Idea™; have'em use FTP instead)

If you are not sure where your ACL privileges come from or what I am talking about, please read the following, in the order presented:
http://magazine.joomla.org/issues/issue-jan-2012/item/637-Joomla-1-6,-1-7,-and-2-5-ACL-Concepts-Overview
http://magazine.joomla.org/issues/issue-feb-2012/item/639-Joomla-ACL-Access-Levels
http://magazine.joomla.org/issues/issue-may-2012/item/761-Joomla-ACL-Configuring-back-end

Akeeba Backup only has a limited interface for front-end backups, suitable even for older generation featurephones. There's also a good reason for this. If you take a backup of the site, you have a complete copy of it. If you download it, you can easily assign yourself as a Super Admin on a locally restored copy of the site and study it. Therefore it would be a blatant security risk allowing front-end users to backup a site. If they have a reason to back the site up they are already Super Administrators and have backend access. As a result the only reason to need a front-end backup is in an emergency situation, most likely not having access to a proper computer/tablet/smartphone or with very limited bandwidth. That's how the front-end Lite mode came to life.

So, you cannot just create a menu item to the front-end Lite mode and expect it to work. Please read the documentation regarding how it works. If you understand it you'll see that you can create a URL type menu item to the front-end mode and give the secret key to trusted users. Be very aware that should the secret key fall into the wrong hands it's trivial for a malicious user to abuse it in order to not only take a backup but also download it to his computer. You've been warned: the front-end end Lite backup mode should only be used in an emergency.