I tried to figure this out years ago and wasn’t able to discover. Please do circle back to this post if you have success.
From my knowledge, there’s no way to export their names/company names. But, if you don’t know this already, you can go to your company’s LinkedIn profile > hit Analytics > Followers, and from there you can select the date range and download an export that includes New Followers, Location, Job Function, Seniority, Industry, and Company Size. Keep in mind though, as I’ve talked with a LinkedIn Support rep, that no matter the date range you select, it will reflect data to-date. I’ve started doing these exports on a monthly basis. Hope this helps!
You can export a list of the connections you have on LinkedIn at any time.
To export LinkedIn connections:
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Click the Me icon at the top of your LinkedIn homepage.
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Select Settings & Privacy from the dropdown.
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Click Data privacy on the left pane.
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Under the How LinkedIn uses your data section, click Get a copy of your data.
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Select Want something in particular? Select the data files you’re most interested in.
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Select Connections.
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Click Request archive.
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Enter your password and click Done.
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You'll receive an email to your Primary Email address which will include a link where you can download your list of connections.