Is it possible to record all the SQL executed by a session? There is a web application which is connected to my Oracle database server via JDBC.

When a user does some actions on that web application, I want to know what SQL statements were submitted to database. Is this possible?
You can find all SQLs that were executed in the sys.v$sqltext view. Together with the sys.v$session view you can see who executed it. Is this what you are looking for?

Otherwise check to see if you can get the information from the application. See if you can change the log level for the classes that access the database.