Can the ICP2(G3) keep the T_VDD output enabled after programming is completed? We need it to continuously supply VDD.

No, ICP2 family programmers always turn off all voltages after programming to prevent hot insertion. However, you can enable power again using specific ICP DLL functions after programming.
Preset D/A of Vdd output: Icp2DaVolt(DA_VDD, )
Release RESET pin:  Icp2PinState (T_VPP, PIN_Z, &dummy).Note: weak pull-down resistors are always connected (300K for I/Os, 160K for VPP)
Turn T_VDD on: Icp2PinState (T_VDD, PIN_HIGH, &dummy)

Turn T_VDD off: Icp2PinState (T_VDD, PIN_LOW, &dummy)
Important: The next programming operation will override previous settings.