At 900, we had the same issue a few years ago. We got so big that spreadsheets just weren't managable.
I developed our own time logging system, and we've open sourced it. Each user gets an ID and they can log in with that. We have a raspberry pi set up with a monitor and barcode scanner, so each of our members swipes a barcode as they come in or leave. You could also use a keyboard and type it in.
The code can be found here:
https://github.com/FRC900/timesheet. You may need to make some modifications to the config files for your team's requirements. If you have any questions, feel free to message me for my email address (I'm more reliable about checking that).