Pet Labor system can be accessed by using the Labor Traveling Machine .
Just use the Labor Traveling Machine and a new option will appear.

– Your pet needs to be level 45 and must have an Intimacy of level 5 before you can let your pet work.
– Your pet must be in Egg State.
– Pet should not be in Battle Mode.

Choose which Location that are available for Labor.
– The one with PLUS sign is one that is available, in my case since I still did not unlock the others, only the item shop is OK for me.
– Yes, you can have the same pets like what you see in the screenshot, I choose to have 2 porings since taming items for porings are not that expensive.

       

You can also check what you can get in that specific work.

After a few hours of Labor, you can see that I have 11 Gifts to received from Item Shop.
Just click it to receive the items.