Apple - Can I record a phone call on an iPhone?

Beware of legal issues.

You can but you'll need to jailbreak, use extra hardware (plug a mic into the headphone jack), or use 3rd party SIP software. For example, Google Voice has an iPhone app, and Google Voice supports call recording. Google Voice does announce to both parties that it is recording the call.

As far as I know 3rd-party, Apple-sanctioned apps - ones you'd find in the App Store, jailbroken apps notwithstanding - do not have access to the phone beyond dialing. In other words you won't find anything in the App Store.