Manual:$wgAllowCopyUploads

Details
Users also need to have the associated user right 'upload_by_url' before this option is available. By default only administrators (sysops) have this permission. To make the option usable from Special:Upload, you need to set $wgCopyUploadsFromSpecialUpload to true as well.

By default any source location is allowed, but you can whitelist a specific set by using wgCopyUploadsDomains

Usage
Add to the bottom of your LocalSettings.php: (The above example will let all autoconfirmed users upload by url).