I purchased a food item (cinnamon tea bag) for a trapped minion, and it is in my inventory, but the site says that I do not have that it in my inventory.

if I had to guess, the minion who was asking for what you wanted was replaced by a different minion before you could feed it, and that different minion was asking for a different food.

I refreshed the page and the same minion was still hungry. I had the item the minion wanted but it didn't register that I really had the item when I tried to feed it.

Yes I did return to see if they still wanted it, and they did. I'm using Firefox. I fed a few minions after this happened and it was ok.