Mulitple switches connected to one pull-up resistor vs separate resistor for each
The decision to use a pullup resistor is more to do with the switches than with the input currents for your device. 
Since you are using a CMOS memory, the input currents for both high and low are very very low (2uA max). This means the maximum input current for all sixteen input to Vcc is about 32uA or less. If you used a 1K Ohm  pullup the voltage drop under worst case conditions would be about 32mV .....totally insignificant in defining the high voltage.

It terms of the switch however, there are other concerns.
If the switch is an overcenter break before make, you are safe, but if it's a make before break configuration  (like many small slide switches) there can be problems. Direct use of VCC (without a pullup resistor as suggested in other answers) is not recommended practice, and could short VCC to ground in the worst case. It is NEVER good practice to use VCC or in fact any voltage supply without defined current limiting as a pullup point. 

My recommendation: Use a single 1k Ohm pullup to VCC as your common pullup point for all sixteen switches.