We use Berkeley Db as a persistent database, which holds data less frequently changed data. Can is it possible to use SQL Developer tool or any other front end tool that can connect and communicate to Berkeley db file for INSERT, UPDATE, DELETE records.
I assume you're referring to Berkeley DB SQL (BDB SQL API).
SQL Developer is not certified to work with BDB SQL; see the SQL Developer Certifications page.
Something similar was asked some time ago, about connecting JDeveloper to BDB SQL; see this Re: How connect Oracle JDeveloper 11g to Berkeley DB 11g. The SQL tools in JDeveloper and SQL Developer are similar and built on the same components, so my answer there can be extended to SQL Developer as well. While you can probably have SQL Developer properly connect to a BDB SQL database, I am afraid you won't be able to do more than just viewing the data; likely any attempt to generate DDL or DML statements, export data, or run DML statements against the database will fail.
If you migrated to BDB SQL from SQLite and were using a front-end tool for SQLite, it should be possible to use that with BDB SQL as well.