Several databases can be used to make a small inventory management system. You can use Kexi leverages SQLite database for each inventory you create.
You can also use the following database when you plan to make an inventory management system no matter if it's small or big:
• MYSQL database
• Microsoft SQL-server database
• Microsoft Access database
• Excel ...
Database and back-end (PHP) hosting tends to not be "forever free" in the same way as you might find things like github pages to host static (no database or backend) sites since databases and backend scripts requires more resources to host and organizations cant give those away for free as easily.
Some of the cheaper solutions like ...