Could someone recommend a db management tool that supports SQLite, MySQL, PostgreSQL, and is web-based?
If a remote desktop is an acceptable web-based solution, you could have Navicat Premium on a remote server that way you don't need the tool locally (premium is the cross-databases edition of the tool: it has MySQL, PostgreSQL, SQL Lite and even SQL Server) .

But you would find cheaper/free if you simply install a tool per database type (e.g. phpMyAdmin, Chive, phpPgAdmin, SQLiteManager).