Is there a way to buy a customized minifig?
There are a few options, however many of them either use standard parts, or require minimum order quantities (Minifigures.com for example requires a minimum run of 50).

A couple of options for bespoke, one-offs seem to be:


minifigs.me - who offer both a Custom Torso option and a Custom Complete Minifigure with printing on the legs and head as well. 
minifigs.com - Offer both torso engraving and printing depending on the complexity of the print.


Of the two, I've seen Minifigs.com at shows and the quality is very good, and the price seems reasonable (especially for the engraved options which don't require a set-up fee).

Alternatively, as suggested in this answer, you could look at doing it yourself with water decals and varnish, depending on your art skills.