LED with variable wavelength
According to https://en.wikipedia.org/wiki/CIELUV#/media/File:CIE_1976_UCS.png you only need a range from about 450 to 540 nm, if you can combine with a 400nm blue and 700nm red (at least if the light is being viewed directly rather than reflected).

Certain types of solid-state lasers have a wavelength that is modulated by a control current, but maybe by 5% (some tens of nm) not almost 2:1. 

You can adjust the wavelength of an LED by changing the temperature, for example, with a Peltier device, but again even 450-540nm, let alone 400-700nm, is not likely possible.

Your best bet is probably a broad-spectrum source such as an incandescent bulb with a monchromatic optical filter.