I need to actually work on my pets instead of loving them from afar, so where better to start than the lovely Kapp'n/ He's gonna be my fan pet for Animal Crossing. I have no idea what to put in his chest other than the Kappa items though. Any ideas from you guys?

There are lots of bugs in this news post! You can catch bugs in AC from what I remember? I haven't played for a long time!
Also:
def need the fruits maybe some of the flowers adding fish/critters to Alex's list
All the fruit, like others mentioned, and definitely the insects!
I like Lantern's flowers, but the old-school ones also look a bit like the ones in the game:
And trees:
Also, maybe some of the deco items from Home Furnishings and Seedlings:
That and my (least) favorite activity, swimming and diving:
And then when the neighbors give you pictures:
Or any of the wrapping paper or stationary: