I came across a forum thread last week asking the above question, "Does a sensor make the camera?" My first thought was a snarky reply like "No the sensor does not make the camera, robots on a production line do.", but decided not to indulge my dark passenger.
So rather, to entertain the question...
The sensor really is a crucial component in the digital photographic process. One could compare the sensor in a digital camera to the tyres on a car. Without tyres one would really struggle to drive around at all.
Better sensors can give you better imaging results just as better tyres could give you better traction on the road, however while it is a crucial component, it is not the only crucial component.
<Enter my shadow self> Photography-forum-trolls often forget that photography is about capturing light. If you put the best tyres on the most powerful car and put it on soft sand then it aint goin' anywhere.
The sensor is only one component required for "getting-the-shot" and when one looks at the greater scheme of things it is probably one of the less significant components. Without good lighting and a decent lens, the sensor doesn't really stand a chance.
Here are some other aspects of the camera's design which can make a difference between getting-the-shot and not-getting-the-shot:
The Autofocus System - Nails focus where you want it quickly
The buffer size - Number of shots before your camera calls "Timeout!"
Ergonomics - Sizing and placement of controls reduces user-fatigue and increases user-efficiency
Customization - The ability to change controls to a layout you prefer
Durability - Operates reliably over years of heavy usage
So my answer to the original question is no, but not a snarky no, a well worded well explained no. The sensor does not make the camera and more importantly the sensor doesn't make the shot!