Despite the presence of many methods, which were created to date for the preparation, functionalization, and vulcanization of polysiloxanes, the development of materials based on silicones remains relevant and requires the introduction of new approaches that combine such modern strategies as atom-economical reactions, refusal to use harmful and dangerous organochlorosilanes, and using the principles of green chemistry and minimizing the use of solvents. In this work, we develop modern approaches to the preparation of both linear and branched polyorganosiloxanes containing azidopropyl functions at the silicon atom. In the first part of the work, it was shown that the proposed method for introducing these groups by catalytic rearrangement with the opening of cyclosiloxane is effective and makes it possible to obtain polysiloxanes with different contents of azidopropyl groups and different molecular weights. The second part of the work demonstrated the possibility of postpolymerization functionalization of such polymers by the mechanism of azide–alkyne cycloaddition under “green” conditions, without using solvents and amines. All of the most important functional fragments were introduced into the structures of organosilicon polymers by a single mechanism, under simple conditions, without the use of expensive catalysts, exposure to irradiation, and hazardous solvents. The combination of simplicity and versatility in the preparation of polysiloxanes with azidopropyl groups with the possibility of performing a click reaction makes it possible to directionally obtain the required materials based on universal raw materials and, as a result, can open up broad prospects for serious development and rethinking of the chemistry of silicones.