Does light with a wavelength on the Planck scale become a self-trapping black hole?

A photon of sufficiently small wavelength would not become a black hole. To see this, consider two observers: one who measures the photon as having a Planck-scale wavelength, and another that is traveling at high speed in the same direction as the photon. This second observer will observe the photon to have a wavelength that is longer than that measured by the first observer due to Doppler shifting. The second observer will conclude that the photon does not have enough energy to create a black hole. All observers will agree on whether a black hole exists or not, so the only consistent conclusion is that no black hole forms.

We can also reason in the reverse. A photon of visible light obviously doesn't have enough energy to create a black hole. Otherwise, light bulbs would be dangerous black hole generators. However, due to Doppler shifting, there is a frame of reference traveling at high speed towards the photon in which that photon has a much larger amount of energy--large enough to create a black hole if that was possible.

Be careful when assigning importance to Planck-scale measurements. We do not know if the Planck units have any physical significance. It is not known if spacetime is continuous or not, and if it's not, we have no reason to think that the "resolution" of spacetime is at all related to the Planck length. Just as a counter example, the Planck mass is about 20 micrograms--a small amount, but one that is handled all the time by pharmacists and far larger than any fundamental particle mass.

I can't improve on Mark's answer, but I would add that while a single ray of light cannot form a black hole, light can form a black hole if multiple rays are focussed onto the same point. This object is known as a Kugelblitz.

The important distinction from the single ray is that in a Kugelblitz the net momentum is zero due to the (approximate) spherical symmetry.