Finding 1st-Degree Connections for Target Accounts in Sales Nav

I've gotten a request to find all contacts that have 1st-degree connections with someone at our company for a set list of ~4k target accounts so our sales team can name-drop the connection. The only way I can determine to do this with Sales Nav is running a separate search for each current employee (there's a feature that would work if everyone had Sales Nav licenses, but alas, 90% of our company is not on Sales Nav). Our Clay CSM recommended using this Apify Actor, but after messing around with it today I don't think there's any way to scrape connections for anyone but yourself (it's based on cookies, not LinkedIn profile URLs). Has anyone done or have any ideas on how to do this successfully with Clay?