All right, how about some words from a luthier?
Yes, shape has some effect on an electric guitar’s tone: remember that although the principle elements of an electric guitar are a (conductive) string and a pickup, other factors (such as materials and shape) affect how the string vibrates, and how the string vibrates is a large aspect of tone.
While I agree with Shadowfyre that pickups are crucial to the tone, they aren’t even close to 98% of what makes an instrument’s tone. You cannot make a Gibson Les Paul sound even remotely like a Fender Telecaster, no matter how much you futz with pickups. The body of an instrument is not just a method for maintaining string-tension and mounting a pickup but integral to the character of the sound it creates.
As has been mentioned, wood choice also has a dramatic affect on a guitar’s tone. Density, elasticity, strength, all affect the the tone and the sustain of an instrument. As a resource, some nice descriptions of woods and their tonal characteristics can be found here.
The reason wood characteristics affect tone is that the string and body interact with each other. Vibrations from the string are passed back and forth to and from the body. and as vibrations are transmitted differently through different substances, the behavior of this feedback system changes, and therefore the tone. Another facet of this is that the pickup itself will also vibrate a bit, and since that magnetic field is now in motion as well as the string passing through it, you get more complex interactions.
It may seem a little strange to suggest this, given that the string touches the body in only 2-3 places at any given time, but consider that it’s this same behaviour that allows any acoustic string instrument to create it’s sound - vibration xmitted to the body by connection to 2 or three points. . .
Finally, the OP: Having established why materials make a difference in the sound of an electric, it’s apparent that an instrument’s shape will also affect how vibrations are affected within the body. Based on the shape, it will be easier for certain frequencies to resonate, while other frequencies will be trapped or dampened. A great big, basically round shape will pad or attenuate different frequencies than a small, (say) spiky shape.
I’m not saying in any way, that the shape dramatically affects the tone, but it’s not insignificant, either. Extreme shapes, for instance, will affect the tone of the instrument.