By far the cheapest way to acquire an axe for spoon carving is to get hold of a small to medium second hand axe or axe head and add a handle and/or regrind the bevels on the blade. Older axes will have been made with better quality metal. Alternatively you can buy a cheap new axe and reprofile the blade and handle as Sean Hellman does here.
If you want to purchase a new mass produced axe, then it is worth looking out for Stubai side axes, with just one side bevelled. The Japanese hand axe sold by Axminster Tools in the UK is reasonably priced tool that allows you to get your hand right up behind the blade for controlled use.
In the next price bracket up are hand forged tools, including carving axes rom Granfors Bruk, axes from Wetterlings (although currently only available from their factory store in Sweden!) or some lovely small axes such as the carving axes by Svante Djarv. Go to the tool suppliers page for lists of companies that sell these axes.
Some members of the Gransfors Bruk family of axes. From the top; the large carving axe, the wildlife hatchet and the small carving axe