Can I draw power for a component while this charger is charging the battery?
LiPo batteries require complex charging patterns to charge the battery quickly and properly. The IC used in that package goes through a preconditioning, fast charge, and then constant voltage mode before it considers charging complete.

Drawing power from the LiPo while charging will interfere with this charging mechanism.  It may work, but it will abuse the LiPo and probably affect its life.  It may overheat.

Normally the charging circuit will power the system directly while also charging the battery.  This can be implemented connecting Vcharger -> diode D1 -> Vin, and Vbatt -> diode D2 -> Vin.  When charging Vcharger > Vbatt, so D2 will be reversed biased and the battery effectively disconnected.  When there is no charger, D1 prevents the battery from energizing the charging circuit which is obviously not a good idea.  If you are concerned about the voltage loss due to the diodes, the concept can be extended such that the diodes control the gates of power MOSFETs instead of directly passing the current.