This is probably caused by an anti-palming feature* with the touchpad drivers. Check for something like “PalmTracking” or “palmcheck” or “disable touchpad while typing”, etc. Here are some images of what it might look like:
The best way to tinker with this, if your driver options allow it, is to reduce the post-typing delay to 1 second or so – that way, the touchpad will come back to life just 1s after the last keystroke, instead of waiting the standard 2-3 seconds.
*Basically, some laptops have touchpads positioned under the N and M keys, which means the touchpad falls right where the user’s right hand palm typically sits. Without this feature, normal typing causes the palm to accidentally trigger the touchpad and produce random clicks. But, as you can see, this feature can also be oversensitive and not let you use the touchpad even after you’re done typing.
A US-standard qwerty keyboard tends to be right-heavy, especially if you have a numeric keypad, meaning the center of the keyboard (between g and h) isn’t necessarily the center of the laptop itself. More thoughtful laptop designs will take this into account and put the touchpad between your two palms rather than the geometric center of the laptop width, but the cheaper laptops will just include the anti-palming driver workaround or ignore the problem altogether. This very message board often sees threads from people annoyed that their touchpad DOESN’T turn off when they’re typing and their palms cause mysterious clicks.