No! It is a source.
This is a complex issue which seems seductively but misleadingly simple, and which so many people have gone astray on.
It seems intuitively obvious that since limestone deposition is removing dissolved inorganic carbon from the ocean, that this should be compensated by one molecule of atmospheric CO2 dissolving in the ocean, but in fact the opposite happens.
The reason is that there is much more dissolved inorganic carbon in the ocean, in the form of bicarbonate ion, than there is CO2 in the atmosphere, and the ocean is a pH buffered system due to dissolution of limestone sediments and also acid base reactions involving weathering of oceanic basalts to clay minerals. So the predominant reaction is:
Ca++ + 2HCO3- = CaCO3 + H2O + CO2
That is to say, in order to preserve pH and charge balance, for each molecule of bicarbonate precipitated as limestone in the ocean, one molecule is released as CO2 to the atmosphere. On a geological time scale, this is the major source of atmospheric CO2 along with volcanic gases. The CO2 dissolves in rain water
H2O + CO2 = H+ + HCO3-
The Hydrogen ion then is neutralized by dissolving limestone on land
2H+ + CaCO3 = Ca++ + 2HCO3-
Or it weathers igneous and metamorphic rock minerals, releasing metal cations and bicarbonate, which flow through rivers into the sea, building up Calcium and bicarbonate until they precipitate out as limestone, completing the cycle.
To put this into perspective, the amount of CO2 released to the atmosphere from all the limestone in coral reefs worldwide every year (which is about half the net storage of limestone in the oceans, at least back when our reefs were alive and growing) is only about one part in one hundred of the amount of the amount we put in the atmosphere every year from burning oil, coal, and natural gas, which shows how severely we have perturbed the natural carbon cycle!
With land vegetation it is a bit different. Plants take up CO2 in photosynthesis and store it in organic carbon. When they are eaten, burned, or rot, the CO2 is returned to the atmosphere. So they are a net sink only if the biomass is increasing, but in fact we are steadily reducing it, adding to the atmosphere. There is around 4 times more carbon in soils than in vegetation, and even more in sediments, so these are the real carbon sinks. Soil carbon is decreasing due to bad land management, but this could easily be reversed if we had the will to do so, and is the most effective way of sequestering carbon. Sediments are a large, but not very efficient sink, as most is oxidized by bacteria back to CO2. Sediments are good carbon sinks only if they are anoxic, and the lack of oxygen needed only covers a tiny part of the ocean (but rapidly increasing with global warming, as it makes oxygen less soluble). But for sediments to become a major sink we would need to turn large parts of the bottom water anoxic, killing the fish and marine invertebrates.