That very much depends on the laser. Most of the approaches I've seen discussed involve multiple 'attacks' on a rock spread out over a considerable period of time. You only fire when the rock is far enough above the horizon (say 25-30% of the time if you have 1 ground laser). And then you want to hit it when it is at some sort of apsis, preferably when it is far away with a relatively slow velocity. The sun has a 24/7 line of sight and is always firing.

Pressure from either sun light or laser is very small, and can be ignored for time intervals smaller than centuries. The method proposed in the article is only about thrust created by vaporization

The sun vaporizes. That's where comets get tails.

Does it? I thought that was heated gasses boiling off.

Boiling off == vaporized (ok pedantically, boiling off -> vaporized, but not the other way around).

The Sun only does that on icy rocks, and when they are near. Lasers can (in theory) do that on icy rocks that are much further away.

In space it often isn't so much boiling (liquid+heat=gas) as sublimation (solid+energy=gas). Sublimation can, weirdly, occur well below the freezing/boiling point. That one lone cosmic ray can heat a single molecule of water enough to cast it into space as a gas ... all in an environment where the average temp is well below freezing points. Temperatures in vacuum are strange.