I bought a "Toyota Key programmer" like this:

http://autotoptech.en.ecplaza.net/sitebuilder/images/toyota1.jpg

You place the eeprom in the socket and a programmed key in the target area to record the chip with the key code. But I want to do the other way, to read the chip and program the key. Is it feasible?

This is doable both ways, I use ZedBull to do some of the Toyotas, putting key data to a TPX chip, I use Carprog or AVDI to do it the other way around.