There are a total of 5 levels in this game with Animals that are not needed for level completion, as such these are the only animals in the game that could be considered missable. Every other animal will be collected while completing the levels. The achievement will unlock after completing the level Guavatron with the sheep in the lorry.
Pepperoni Palace has a Turtle running around the level.
The Hoop House has a Chicken in the dog (chicken?) house in the back garden
Case de Cliff has a Goose hiding in the ground floor of the house - it will come out when the mailbox is slapped.
Packmore River has a Frog - You CANNOT pick the frog up, but he follows the closest player to him.
Guavatron has a Sheep, it's on the left-hand side near the TV/big rock.
Hoop House - Moving In Mode Has a Chicken, Thanks to Belvega for this added information.
You need to make sure the animal you're trying to deliver is in the lorry before you put the final item to be delivered in the lorry. All animals (except the frog) can be carried like a package using
or
EDIT 1 - To clarify, there are 7 animals needed for this achievement, the problem is, the spreadsheet I created to track animals/toilets/consoles only has these 5 animals on it, and it's been ages since I unlocked this. If anyone else notices the tracker going up for an animal that isn't on this list if you leave a comment I'll gladly add it to the list.
EDIT 2 - The animal that most people seem to be having issue with is the goose, if you're at 86% there's a good chance the goose is the problem.
EDIT 3 - Added the Chicken from 'Moving In' mode on Hoop House is required.