Probably you had this thought because it's what you've often heard of people doing?
Yes, a full sized tree-cutting-down axe is rather big to be accurate, but a "little axe" as you say, or what people would call a hatchet, is a
very, very common way of dispatching a chicken.
Sharpen it well, practice beforehand if you are not comfortable and accurate with an "axe", I recommend the two nails method, and make sure your swing has a goodly amount of resolution behind it.
Remember that, like pretty much any other method of killing, the body will flap like crazy for a minute or so afterwards, so it is smart to have the wings well wrapped against the body and/or put the body head-down in a narrow wastebasket or such til it is limp again, to avoid blood everywhere and broken bones on the chicken.
Good luck,
Pat