Some keyboard programs on iPhone record your keystrokes locally (on your device). A common reason for this is to improve the accuracy of spelling corrections. Some keyboard programs transmit your keystrokes to some other computer for analysis.
IPAChartBoard remembers what keys you have used, so that it can show you those keys in its "frequently used symbols" (the 🕘 symbol). IPAChartBoard stores this information on your device. It does not transmit any information off your device.
IPAChartBoard will not ask you for "full access."