As you have correctly noted, without the agent process, you will need something else external to the server, perhaps a service you write and install or Windows scheduler.
Note that with an Express installation for a local application, it is possible that the machine may not be on at the time you want to truncate the table (say you set it to truncate every night at midnight, but the user never has his machine on).
So your scheduled task is never run and your audit log gets out of control (this is a problem with SQL Server Agent as well, but one would assume that a real server would be running non-stop). A better strategy if this situation fits yours might be to have the application do it on demand when it detects that it has been more than X days since truncation or whatever your operation is.
Another thing to look at is if you are talking about a Web Application, there might be time when the application is loaded, and the operation could be done when that event fires.
As mentioned in the comment, there is sp_procoption - this could allow your SP to run each time the engine is started - the drawbacks with this method are that for long-running instances, there might be a long time between calls, and it still has issues if the engine is not running at the times you need the operation to be done.