The most likely causes are the lens itself or a body/lens communication issue, not necessarily the D90’s AF motor alone.

Best next step: isolate the problem. Use the Nikon 18-105 enough to see whether the same freeze happens. If it only happens with the Sigma 70-200mm, that strongly points to the Sigma lens. If it happens with multiple lenses, the camera body becomes the more likely culprit.

Because autofocus returns after a power cycle, a contact/CPU communication problem between lens and body is also plausible. A simple test is to remove and remount the lens (or slightly disengage and reconnect it) when the problem occurs, to see whether that restores AF without powering off.

There are anecdotal reports of similar behavior with some Sigma lenses, including this type of 70-200mm, so if your testing points to that lens, sending the lens to Sigma under warranty is the most sensible first move.

Also check whether any custom button setup (such as AE-L/AF-L behavior) could be interfering, though the main priority is to determine whether the fault follows the lens or the body.