Just found this. Pretty silly how they got rid of the "copy emblem" button just to leave this idiocy in.
1. Go to your target's profile
2. Click their emblem and a box should pop up with a flat image of their emblem inside of it
3. Right click the image and click "view image"
4. In the address bar, copy the longest string of numbers.
a. Ex: If the url were "http://eaassets-a.akamaihd.net/battlelog/prod/emblem/796/651/256/2955060247687647828.png?v=1383524377", you would copy "2955060247687647828"
5. Go to your emblem editor at your profile.
6. Paste the string of numbers in your address bar after "http://battlelog.battlefield.com/bf4/emblem/edit/personal/".
a. Ex: If your string was "2955060247687647828", your link would be "http://battlelog.battlefield.com/bf4/emblem/edit/personal/2955060247687647828"
7. You will now be able to see all of your target's emblems.
8. Export the emblems from the current page and then import the emblems into your own emblem customize page by using this method: Exporting / Importing Emblems: The most effective way to share your emblems. : bf4emblems
a. Make sure to use ctrl shift k to open the console.
If you need to see the process, go to this video I just made. It shows up to step 7.
https://www.youtube.com/watch?v=1NIpTt7QwE0