There will not be any locking issues.
I don't know about the exact performance impact. As discussed in that article, SQL Audit is based on Extended Events, but I don't know exactly how SQL Audit uses Extended Events. What I do know is that if you have a workload that runs a lot of short quick statements, capturing sql_statement_completed in an true Extended Events session can cause a noticeable overhead, even if you have a good filter. But the events that SQL Audit uses may be more target and imply less overhead.
I would suggest that you implement SQL Audit, and pay attention to if there are any complaints in the next few days.
Also note that SQL Audit has a few options for controlling how to write to the file, which may be important at a busy site.
I should add the disclaimer that I have not used SQL Audit a lot, but I'm just applying some common sense.