How is the amplitude of light related to number of photons?

Yeah, pretty much. When you increase the amplitude of a light wave, you are essentially just sending more photons of the same kind. The energy of each photon is $hf$ (though with some allowance for wavepackets, where each photon will come with as a probability distribution over a range of frequencies), so if you increase the energy flux you need to increase the photon flux.

When you actually dig down on it things are not quite as simple, of course, because the exact value of the electric field is a quantum mechanical variable analogous to the position variable of a quantum harmonic oscillator, and you can't really describe it as a variable with a well-defined oscillation. (In particular, for example, if you know precisely the intensity of the light, then you lose all information about the phase of the oscillations, and vice versa.) However, the electric field is essentially confined to a region that grows as the square root of the number of photons in the mode.