Step #1: Click the white space in the comment box or reply button first in order for the "Add Media" to show up.
Step #2: Click "Add Media"
Step #3: Go to the right side (where the media section pops up) and click any emoticons you want.
Step #4: Search "Puppy Emoji" for shortcut quick list. You can also search it using the title or if you want, my username for random emoticons and they will be there. Click your choice and enjoy~
Note: Since there's a limit of 20-24 in the emoticon search, I will include another batch with V 1, V 2, V 3, and so forth so that you can search them all
No "Add Media" Issue
I don't really know what's causing the problem but according to dA Help Desk, this problem may be caused by extensions or add-ons from your browser. Take a look at the link which ones are causing issues, thank you~
Got permission to expand them and easy accessible, follow the site to support more of these~