In a nutshell, you can't. As @iant says, WMS is just imagery that is designed to be displayed on your screen. To get the actual data, you'd need a WCS. If USGS doesn't supply that, then your options are limited.
You could make a guess at a pixel's height value if you know the range over which the greyscale covers. For instance, it might be [0-1000] metres, which means that each shade of grey in a 256-colour image covers a height of 1000/256 ~= 4 metres. So you'd just need to multiply the grey value you get by that number to get an approximate height. But unless your horizontal resolution is quite large, a 4 metre vertical resolution would be a noticeable stair-step.
Also, as WMS is designed to produce pretty maps rather than analysable data, you are likely going to get all sorts of other cruft such as place names, roads, and contour lines, that will throw your calculations right off.
If I were you, I'd investigate where they got their source data from to generate these WMS tiles. If its fairly low resolution, it's probably based on SRTM data, which you can get for free for (almost) anywhere on the earth.
Either that, or petition the USGS to provide a WCS server of the data you want, but I wouldn't hold your breath!