The amount of sunlight available should depend on the size of the area they cover. What area of sea floor would they need to cover to produce an amount that could build the chalk cliffs or all the chalk formations on earth?
Lots. Have a look at this map for starters.
The green bits are calcareous ooze. Let's call that about 10% of the sea floor. A quick look round the Internet shows that these deposits vary in thickness from tens to hundreds of meters. Let's call it an average of 25 meters, which is extremely conservative. If all that was spread out on the ocean floor, then, that would be a thickness of 2.5 meters. Now based on sediment traps, the rate of deposition is between 1 and 5
centimeters per thousand years. Call it 5 everywhere (again, favoring you rather than me). So being generous all round, that would be 50,000 year's worth of sediment
if the whole of the ocean surface was dedicated to the production of coccolithophores and foraminiferans, which it isn't. That's
before we get onto the actual rocks, and before we consider whether even a
magic flood could sweep up the ooze into neat zones which just happen to correlate perfectly with the places that the calcareous-ooze-forming organisms happen to live in at present.