The wavelength of the photons emitted by an LED is defined by the bandgap of the semiconductor junction; that's a material composition constant. It can't be adjusted, far as I can see, by means of anything but a temperature change which probably isn't feasible for an LED potted in plastics connected with solder.
So, nope, you won't get a variable-wavelength LED: LEDs are, by principle, fixed wavelength.
If you need an adjustable range of wavelengths: Get a source of broadband radiation (e.g. a broadband IR LED, which is essentially a pretty impure LED, or, even simpler, an incandescent bulb), get a prism adequate to diffract a beam of that light (think making an infrared rainbow) and a mechanism that rotates that prism so that you can select the wavelength you desire with e.g. a slot (followed, typically, by some focussing optics).
There's tunable lasers but I'm more or less certain these are out of your price range of interest.