Can the ADC-VDB780B's LED stay illuminated?

No. The ADC-VDB780B is battery-powered and the constant draw of power from a continuously lit LED would require the doorbell camera to be recharged more frequently. 

The ADC-VDB780B's LED will only illuminate when the button is pressed or when motion is detected. For more information about the ADC-VDB780B's LED colors, see Alarm.com Wireless Video Doorbell (ADC-VDB780B) - Installation Guide.