This is the page in the serial communications protocol manual for the E5AX temperature controller that details the signal connections. Since pins 2 & 3 are used it appears that a standard cheap adapter available on ebay will do the transition from 9 pin to 25 pin rs-232. This should be carefully reviewed but at first review it appears to be the case.
Note that Transmit Data on the computer serial port must be connect to Receive Data on the Omron E5AX controller port. Thus crossover is probably not needed. Refer to connection tables below to verify this. The easiest way to find out is just to connect what you have in hand and see if it works. Report back results and I will update this page so in future this page will contain the correct information.
Port on back of computer looks like this:
Port on E5AX with 25 pin serial port looks like this