If light has energy but no mass, how can it have momentum?
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Because the common-use equation are not complete.
E=mc^2 is the Mass–energy equivalence valid in rest frames. For object in motion it is E^2 = p^2 c^2 + m^2 c^4
The momentum equation p= m v is one of Newtonian mechanics, m is the rest mass. If you include the relativistic effect that is important when speeds start to approach it is the speed of light m is the relativistic mass, that is the total energy of a body divided by the speed of light squared
At a speed that is common for objects with a mass on Earth the difference between rest mass and relativistic mass is so thin that we can simply ignore it.
For a massless particle, you can use the Mass–energy equivalence and derive that momentum p = E/c where e is the energy and c is the speed of light
https://en.wikipedia.org/wiki/Mass_in_special_relativity
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