There is no reason you could not, if you followed the standard requirements for outdoor recepticles, such as
- Ground Fault Circuit Interrupter (GFCI)
- sealed box
- properly protected cable
However, this is likely to be a bulky add on.
Cutting a hole in the post presents a problem becasue you need to install a sealable box and sealable cable connections.
Finding a post that already has a UL listed outlet is probably a safer, easier and maybe cheaper alternative. (See comment by @The Evil Greebo) A manufacturer can design an integrated housing for the outlet and get it UL rated, whereas you need to rely on off-the-shelf parts (read not custom fit) to get the same safety and performance.
Also, on the posts you have seen so far, it might be easier to remove the decorative items and plug them with small caps or finials than adding a safe outlet.