Are your devices managed? If not, that should be step 1. That should help build a bulk of your inventory. If they're already managed (or some/most), you'll probably need to clean up some of the ones that haven't checked in for a while, then hunt them down manually.

Conditional access would help a bit here as well to help make sure everyone is on a managed device. Notify, notify, notify that resources x,y,z will only be accessible on a managed device starting on x date. If someone can't work you'll be hearing from them. Obviously this requires buy in from management, but shouldn't be a hard sell since you're killing 2 birds with 1 stone.

To find whatever remaining devices are out there, work with finance. Pull invoices, emails, orders, whatever to try and track stuff down. Also, don't be afraid to reach out to users. "Hey John, I saw you got this laptop originally, then you got a replacement, what happened to the old one?". It's manual and it sucks but devices for 200 users isn't that bad.

Last but not least, once you're done this exercise, develop policies/processes for hardware/software procurement and asset management. The sooner you get this under control the easier it will be, but you need to keep on top of it. I mention software as well because if you're planning on being there a while/the company is growing, I guarantee you that will come up in the future.

There's a million ways to go about this, with varying levels of cost. These are just a few ways that I've found successful in the past.