No, the audit trail does not show the passwords of failed log in attempts. Using the audit trail you can determine things such as the machine name and OS user that is attempting to log in, but it will not capture the password that was used.
Since this application has been logging into the database for years, I am guessing that the database itself has been updated a number of times, correct? If so, is it possible that you are running into password case sensitivity issues as a result of an older client being used on the application side? If the application is connecting from a machine using an older client, you may want to check out this question: ORA-01017 Invalid Username/Password when connecting to 11g database from 9i client
To test this hypothesis, you could try logging in from the application side by using quotes to pass the credentials. From the question above:
oracle9i defaults to uppercase as it didn't cater for case
sensitivity. instead of changing the database to insensitive, you can
connect by pasting your password in double quotes eg `sqlplus
youruser/"Password"@db to pass mixed case.
If that is not the case, and you want to determine what password is being passed from the application to rule out that avenue, you could use a tool like Wireshark to listen to the traffic and see if you can sniff out what credentials they're using, assuming they aren't using an encrypted connection. Please only explore this if you have complete control over the network or the permission from the appropriate powers that be, as it could be illegal or against your organizations policy to use a tool like Wireshark without permission.