You ask: "At steady state, does a full fridge require more energy input to keep cool compared to an empty fridge?"

Generally not. It does to some extent depend on the nature of the load. If there is a significant convection current in an empty fridge, which generally there should be for efficient cooling, then we will constantly have the maximum of cold air washing against the insulated walls, which would imply the maximal loss of coolth (read, maximal inflow of heat) through the insulated walls. 

A full fridge will interfere with convection, thereby reducing the burden on the insulation. The items nearest the outer walls will be subject to a heavier influx of heat, as compared to those deeper inside, and thick items will absorbe more heat before passing it on to the cooling coils etc, but the overall flow of heat towards the cooling components will be hampered and reduced, reducing the loss of coolth at the cost of less effective cooling near the walls and disproportionately effective cooling of the deeper-seated objects.

YMMV of course. The ultimate cooling load will be a function of the temperature gradient between the inner and outer surfaces of the insulation.  Some kinds of loads might affect that gradient more than others.

But let's re-emphasise: You did say: "At steady state..." I ignored transients at all points of the argument, and transients generally are very important in such matters.

At a steady state, I assume that the entire contents of the refrigerator are at the required temperature. The only source of heat is through the walls of the refrigerator  and so the inflow of heat should be the same empty or full. The energy required to keep it cool is therefore the same in both cases.

I note what Jon says about convection currents but my fridge has a fan that circulates the air and so should eliminate this problem. It is, in fact, a side by side design with a single cooling coil and two fans that blow cold air around the freezer and/or refrigerator as required.

The advantage of a full refrigerator comes when the door is opened. An empty refrigerator will dump a large volume of air out and take in the same volume of warmer air from outside. A full refrigerator will dump a smaller volume of air and take in less warm air. The full refrigerator will therefore have to pump out less energy to get back to a steady state.

I can't believe that it costs the same to keep a greater amount of stuff cool. I don't reallty have the nouse to back this up, but here is my best guess.

They are 3 types of warming that I know about, convection, radiation and the one where molcules excite each other (forgotten the name of this one).

A fridge should be airtight so convection shouldn't be an issue.

Fridges presumably have some insulation to stop radiating heat but this won't be perfect, so a larger mass inside the fridge will be warmed more than a lower one.

Air molecules, being quite spaced out, find it quite difficult to pass on their jiggly energy to each other, which is why air is a good insulator (convection aside). Denser materials are better at this. So if the inside of the fridge is fuller the molecules in it will be excited more easily and it will conduct heat in better.