Yes, if a light wave starts orbiting a black hole, both its frequency and wavelength will change. This phenomenon is known as gravitational redshift.
As an object, including light, approaches a massive object like a black hole, it experiences the gravitational pull of that object. The intense gravitational field near a black hole causes space to be curved, which affects the properties of light passing through it.
When a light wave moves closer to a black hole, it has to overcome the gravitational pull of the black hole, which requires energy. As a result, the energy of the light wave decreases, and according to the wave-particle duality of light, the frequency of the wave decreases as well. Since the speed of light is constant in a vacuum, the decrease in frequency leads to an increase in wavelength.
This shift towards longer wavelengths is called gravitational redshift because it causes the light to be "redshifted" towards the red end of the electromagnetic spectrum. The amount of redshift depends on the strength of the gravitational field near the black hole and the distance from the black hole.
Therefore, if a light wave were to orbit a black hole, its frequency and wavelength would change due to the gravitational effects, resulting in a gravitational redshift.