Why not create your database in Microsoft Access? The Jet MDB drivers are going to work great with Excel and be installed as part of Office.
Microsoft has spent years trying to get a shared file database to work on a LAN. In short, it's not possible, but they've managed to get it to work nearly 99% of the time, and they have tools for fixing it when it breaks. SQLite is designed as an embedded databases. While it does attempt to do file locks, it's not designed to be a great multi-user system. Like all shared file databases, it too can have problems with multiple users changing data at the same time. Additionally, the Jet (mdb) database has the added feature of being very easy to later upgrade to a Microsoft SQL Server - either their embedded/free version, or a full licensed server. You can't one click upgrade SQLite in the same way.
With any database, make sure you make a nightly backup part of your plan.