Record video from webcam w/o microcontroller
Webcams have video compression and USB protocol "baked in" to the integrated circuits it is made from.  That means the video is not accessible unless you are speaking the proper USB protocol, and it takes a microcontroller or microprocessor or computer to do that.

There are very tiny video cameras that have actual video output (vs converted to USB) and those are suitable to connect to a transmitter so you can view/record the video back on the ground.  There are even very tiny video cameras with built-in recording to a micro SDHC card, etc.  Those would seem more suitable for use in a mobile device like a rocket, drone, aircraft, vehicle, boat, submarine, animal, etc.

USB "webcams" are not suitable for the kind of application you are asking about. Virtually all of the cameras built into phones and laptop computers actually connect to the processor via an internal USB connection.  And the ones that don't likely have a more complex interface that would be difficult to deal with in something small and light enough to go aloft in a rocket.