A better way of stating the claim is as Dr. Galvin A. Schmitt does on this NASA site:

as temperature rises, the maximum sustainable water vapor concentration increases by about 7% per degree Celsius

Then, look at a water vapor pressure table

Near 0 degrees C, the per cent increase per degree is about 8%.

At 25 degrees C, the increase is about 6% per degree.

No, 7% is somewhat too high.

According to Recent Climatology, Variability, and Trends in Global Surface Humidity Journal of Climate, (2006) volume 19, pages 3589-3606:

The observed dq/dT is about 0.58, 0.38, and 0.77 g kg^-1 °C^-1(r² = 0.81, 0.77, and 0.76) for annual q and T for the globe, global land, and ocean, respectively. In percentage terms, they are about 4.9%, 4.3%, and 5.7% change in q per 1°C warming, which are close to those (∼5.4%, 5.1%, and 5.5% per 1°C, respectively) suggested by the Clausius–Clapeyron equation or its empirical version [Eq. (3)] for saturation specific humidity (computed locally and then area averaged). ¹

where "q" is specific humidity, "T" is temperature, and "dq/dT" is the rate of change of specific humidity with respect to temperature.

and where footnote 1 is:

The Clausius–Clapeyron equation locally gives 6.2%–6.4% change per 1°C in surface saturation specific humidity (q_s) for air temperature within 15°–20°C. For regional averages, however, this percentage is lower because the area-averaged mean q_s is higher than that calculated using the Clausius–Clapeyron equation and area-averaged mean air temperature and pressure.

So, in conclusion, the correct value is 4.9%.