Can the VPD1 and VPD2 power control pins on the HMC414MS8GE be connected together and controlled with a single voltage source, such as a potentiometer or DAC? What are the implications if the voltages are not changed simultaneously?

VPD1 and VPD2 should always be connected together and driven in parallel. They should be driven from a voltage source. That voltage can then be adjusted dynamically. There is a plot on page 4 (11-61) of the datasheet that shows how gain, current, P1dB, and PSAT are affected by the VPD1/2 voltages. While a potentiometer can be used, it has not been tested for reliable operation. Based on the typical VPD current of 7mA at 3.6V with a 5V VDD, a resistance value of 200 ohms might be a reasonable starting point. However, it is recommended to drive VPD with a DAC if you want to vary the voltage.