Yes, Lightroom can do this with some limitations. Most likely it is not the solution you need. First, employees will need to sign into a single account. Adobe limits two devices at a time be signed into a single account. You could work around this by using a mobile device, because there are no limits for mobile devices. This probably isn't a viable solution though.
Second, how large is your company's photo collection? You can purchase additional Cloud storage from Adobe, but I hear it is expensive relative to other services.
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