Would it be feasible to build an RC plane using the entire wing as an
elevon, with each wing independently moved up and down with servos?
Yes, such designs exist. They are most often seen in radio-controlled slope gliders. The key is that there is also a horizontal tail which is fixed. Otherwise there would be no pitch control-- moving the wings together in the same direction would accomplish nothing in a tailless design.1
If the wings move in the opposite direction for roll control, but pitch control is accomplished with elevators on the horizontal tail (or with an all-moving horizontal tail), then the design is described as a "wingeron" design. (The term is a "portmanteau" of "wing" and "aileron".) With proper gearing, "wingeron" designs can be built with only one servo driving the wing movement, or two servos can be used.
If the wings move in opposite directions for roll control and also move together for pitch control, as posited by the O.P. here, then the aircraft is described as a "pitcheron" design. This word is "portmanteau" of "pitch" and "wingeron". "Pitcheron" designs always need two servos to control the wings.
Here is a photo of a "Pica" radio-controlled slope-soaring glider-- a "pitcheron" design, with no elevators or other moving parts on the horizontal tail (actually a V-tail in this particular case). (Source: "Slopeflyer" website )
Here is a photo of a Sig "Samurai" radio-controlled slope-soaring glider-- a "wingeron" design, with elevators on the horizontal tail (again, actually a V-tail in this case). (Source: "Wanderings" website.)
Note the absence of ailerons in all the photos.
Here's a link to a YouTube video of a Sig Samurai "wingeron" glider in action. The internal workings are briefly visible near the start of the video. One single servo controls both wings for roll control. One more servo to control the elevator function of the tail would have been sufficient. In this particular case the glider is set up with a second servo for the tail (making three servos in all) to allow the moving surfaces to act as "ruddervators" (as in a Beechcraft Bonanza) rather than just as elevators, but the rudder function is really not necessary in a fast aircraft of this nature, and certainly is not a significant source of roll control, since the wing has neither dihedral nor sweep to create slip-roll coupling. As noted in the video, this particular design is often built with just two servos, omitting the rudder function of the tail surfaces.
Link to a related question on a model airplane discussion forum: Pitcheron & Wingeron. What's the difference?
Note than in the case of a powered aircraft, a "pitcheron" design will create a "thrust vectoring" effect that might be less than optimal, depending on the specific details of the design. When the pilot increases the angle-of-attack of the wing by moving the control stick aft, he is essentially tilting the thrust vector slightly downwards relative to the wing, which seems less than ideal. However, if the thrust line can be configured to pass through or near the CG of the aircraft regardless of wing position, then this is not the same as an increase in "downthrust" angle, which would generally have much a more unfavorable effect on handling. Naturally, you could always consider mounting the motors on the wings.
Finally, note that a key feature that allows the pitcheron or wingeron designs to work well in radio-controlled models is that there is no need to consider the aerodynamic "feel" transmitted from the control surfaces back to the control stick, or the position that the controls will go to if the pilot relaxes all forces on the control stick and just lets the control stick "float" freely. In radio-controlled models, we are only concerned about stick-fixed stability, not stick-free stability. Likewise we are only concerned with the control surface position that must be maintained to obtain a given result, not the force that must be applied to the control system to obtain that result. The situation is very different with full-sized airplanes, at least so long as the control system is "reversible", i.e. so long as aerodynamic forces can be transmitted from the control surfaces back to the control stick or yoke. The same difference also applies to all-moving horizontal or vertical tails-- they are simpler to implement in radio-controlled models than in full-scale aircraft with conventional "reversible" control systems.
- Re "moving the wings together in the same direction would accomplish nothing in a tailless design" -- except for hang gliders and other weight-shift-controlled aircraft! The key here, of course, is that if the CG is far enough below the rotation point of the wing (which in a hang glider is essentially the point where the pilot's "hang strap" is attached to the rigid structure of the glider), then rotating the wing (relative to the rest of the aircraft "structure", i.e. the pilot's body, or the "trike" unit in the case of a powered "trike") accomplishes a significant forward or aft translation of the CG of the whole system, relative to the wing.