Calcareous biominerals typically feature a hybrid nanogranular structure consisting of calcium carbonate nanograins coated with organic matrices. This nanogranular organisation has a beneficial effect on the functionality of these bioceramics. In this feasibility study, we successfully employed a flow-chemistry approach to precipitate Mg-doped amorphous calcium carbonate particles functionalized by negatively charged polyelectrolytes—either polyacrylates (PAA) or polystyrene sulfonate (PSS). We demonstrate that the rate of Mg incorporation and, thus, the ratio of the Mg dopant to calcium in the precipitated amorphous calcium carbonate (ACC), is flow rate dependent. In the case of the PAA-functionalized Mg-doped ACC, we further observed a weak flow rate dependence concerning the hydration state of the precipitate, which we attribute to incorporated PAA acting as a water sorbent; a behaviour which is not present in experiments with PSS and without a polymer. Thus, polymer-dependent phenomena can affect flow-chemistry approaches, that is, in syntheses of functionally graded materials by layer-deposition processes.