Use touchpad while "typing"?

This is called PalmCheck. To disable this, go to Mouse Properties in Control Panel, select the Device Settings tab, select the Synaptics Clickpad or similar device in the list and click Settings, and under PalmCheck in PalmCheck-Enhanced, drag the slider down towards Off.

Since disabling SmartSense didn't solve the problem for me, I had to dig a little deeper.
After much toil and bickering with various forms of Sony Support, I gave up and tried messing with the Registry. After some failed attempts to disable the Synaptics filter driver that's attached to the keyboard, I found something that worked.

Solution:
In the Registry, go to the key HKEY_LOCAL_MACHINE\SOFTWARE\Synaptics\SynTP\Defaults, and set the value of all string entries of the form PalmKms…, to 0. (Of course, they probably don't all need to be 0, but have fun finding out which one(s) you actually need!)
These values determine the amount of ms that the touchpad stays disabled after keyboard input has been detected.

I think Sony will probably issue an update to the VAIO Control Center where you can simply disable this functionality, but until that day, this fixes the issue.

In Windows 8 the option is called SmartSense. Disabling this worked for me.

1. Press Win+W to bring up Settings Search
2. Type "synaptic"
3. Double Click "SmartSense"
4. Slide the Slider all the way to "Off"
5. Click "Close" and then "OK"

This will Disable the Anti-Palm detection which prevents simultaneous Keyboard and Mouse use. I recommend you re-enable this feature when you're not gaming as otherwise you risk problems while typing.