Photogrammetry is a well established technique for doing this.
I just did a Google search with "wave height measurement photogrammetry" and got a large number of hits. I downloaded 2:
1) CLOSE RANGE PHOTOGRAMMETRIC MAPPING OF SEA WAVES FOR BETTER
HARBOR DESIGN
Hatzopoulos and Demetrios, ASPRS annual conference 2012
2) Using stereo photogrammetry to measure coastal waves.
S. de Vries et al, Journal of Costal Research, SI 56, 1484 1488, 2009
3) Also note that if you are able to float any material on the surface of the water that will make it relatively easy to do a stero or multi photogrammetric measurement of the surface in 3D and in a time sequence if required.
As Stephen mentions, photogrammetry is a good technique. An alternative would be a fast rotating laser scanner system, though this would be better at getting sequential 2D slices of data rather than capturing the whole 3D surface in an instant.
Photogrammetry is exact subject I have found. Actually, I have a program, "Photomodeler" and this program seems to adopt the principle of photogrammetry. I`ll investigate this topic.
I don't have personal experience of Photomodeler but it generally gets good reports and seems to be a useful tool for both experts and non-experts. Have fun.