Write to the firm without disclosing what you have heard or that you know there is an opportunity. This does not say there are vacancies. It means no more than that a company employs people of your kind. You can add to this list firms or organisations found in your own research. To all of them you could make speculative or 'cold canvass' approaches.
Cold canvass is now much more widely practiced than it used to be.
Job hunters at all levels and in all disciplines have found it produces results. Personnel managers and others have thus become used to receiving letters and CVs and may not always take the trouble to reply when they have nothing to offer. This is no reason for abandoning the idea, but it does mean that every letter must be prepared with meticulous care to make any impact. It should show that homework has been done and that writers have no doubt of their ability to give a valuable service in and to the firm. Your network may have given you leads, but everything needs to be checked.