Errr... as for "making a post", if talking about in the forum, I don't believe you have the ability to upload to the gallery from there (but you can embed from existing images from any gallery you have uploaded images to).
As for the gallery itself, if you don't have the correct permission set to add albums then you will not see it (Admin normally has all permissions).
You specifically need (for that user group) to make sure that your settings in the XenForo Medial Gallery Album Permissions are set appropriately for that user group.
Suffice to say, it does work (as the screenie I attached is from a normal user account on one of my forums).
Unluckily more detailed help is not normally provided in the public area like this but in the licensed holder area.