It's not replicable in real life
As AndreiROM said in his answer, it is extremely difficult to edit millions of cells' DNA precisely, and that shape a developing fetus - not a living human. Even with genetic engineer viruses, you cannot change the physical structure of a human being rapidly and at will.
Here's an alternative (with handwaving)
Prostheses (plural of prosthetic) work well in real life to change the appearance of the face - but carrying around a kit with latex and makeup just isn't "cool", however realistic. Using a combination of strong quaternary ammonium cations (shown safe for use on flesh here and here) to soften tissue substantially, and whey protein (info here) for muscular rigitity, you can create a substance that makes skin and muscle ~somewhat~ posable. It would be applied in a mask, through facial injection, or through a topical cream, and it may allow your spy to shape their face to a small extent.
After some time in art school, Clayface may be able to work with this.
A stretch, and it may require further development of chemicals, but organic prosthesis is more realistic than shapeshifting on demand.
Or if you'd rather be realistic
Consider normal, liquid latex prostheses and makeup, which are proven to work just fine for "impersonation". Not as exciting as the above method, but much more realistic.
Side effects (for 1st alternative method)
May increase cause cell death over time due to stress; may increase rate of cell death due to antibacterial properties; may tingle or sting slightly.