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?

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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.

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HMC414MS8GE?, HMC414: FAQ