The NI 9481 is rated to 2 A at 250 VAC. Not quite enough for your needs.
I prefer not to bring kW level power to the computer/DAQ equipment. I would use a Digital Output device to control a solid state relay. Most Digital Output devices can drive SSRs. Check to see if the Analog Input device you will be using for monitoring the oven has Digital Outputs which can do the job.
Another approach is to use an off-the-shelf temperature controller with RS-232 or Ethernet connections. This has the advantage of the built-in open thermocouple protection and PID autotuning which many controllers have. Even if the computer crashes the temperature controller probably will just continue in a safe mode.