Actually the answer is yes. Here is an example how you can do this: http://html5doctor.com/introducing-web-sql-databases/
The bad thing is that it’s with very limited support by the browsers.
More information here HTML5 IndexedDB, Web SQL Database and browser wars
PS: As @Christoph said Web SQL is no longer in active maintenance and the Web Applications Working Group does not intend to maintain it further so look here https://developer.mozilla.org/en-US/docs/IndexedDB.
SQL.js
EDIT
As @clentfort said, you can access SQLite database with client-side JavaScript by using SQL.js.