Hi @wistranderic,
I’m afraid that is how WordPress works. It prevents any users with no privileges from uploading an image or attachment. If you want to post an image, you need to use an image field and set uploader' => 'basic'
in your acf_form(). This page should give you more idea about it: http://www.advancedcustomfields.com/resources/acf_form/.
I hope this makes sense.