Despite what the achievement seems to want to require, this does not count any photos taken with the instant print camera. You must take 5 photos with the other camera.
You need to reach reputation level 3, which will unlock the better camera. You can then buy this from the store. Once it arrives, simply take 5 pictures (doesn't have to be of any pets or anything, just spam the trigger) and the achievement will pop.
Edit: After further testing for the 50 photos achievement, that is able to unlock by taking pictures with the instant print camera as long as they DON'T include an animal in them, so it's possible this will be the same. If someone tests this please let me know if it works and I'll update the solution.