Seems to be a logic level part
Will have at least 1V drop across it, see Figure 2.
TPIC6B595 will have 3-4 ohms of resistance, see Figure 7, at 20mA = another 0.07V
So as long as your LEDs have Vf <(5-1-.07) = 3.93V you should be okay.
Adust your current limit resistors accordingly.