Shaking electric furnace
The blower fan should not be running continuously. You must get that repaired. Is the switch on the thermostat set to auto or to run? The thermostat can be incorrected wired in a way that keeps the blower fan on continuously.

The filter below the fan may be too restrictive or the ducting leading to the filter is constricted. Put in a new filter of the proper type. Some filters are too restrictive for a given furnace. 

But note that these fans must have a certain load on them supplied by suction on the back side and/or high pressure on the outlet side from the combined resistance of the heat exchanger, the evaporator coil, and the ducting. If they are run without sufficient load, the motor will over-speed and burn out.

The quieting when you open the access panel might be due to the fan speeding up and getting out of a resonance that is causing vibration. Also the bearings on the fan motor might be bad or need lubrication. Some of these fan motors used to  have bearings which required oiling through channels but most nowadays are permanently lubricated.