Is it possible to make fudgy brownies without oil?
If you have a recipe that already produces a dense fudgy brownie and you're looking to do it without oil, you're just out of luck.  

If you're looking to produce a dense fudgy brownie and wondered if, while you were at it, you might cut back on the oil...NO, but here are some suggestions for making a cakey brownie more dense and "fudgey":


Follow Tim's advice above on using butter instead of oil.  It's not doing anything as far as calories but it will produce better flavor vs. a neutral oil.  However, particularly if you're planning to put nuts in them, you could replace a portion of the vegetable oil with walnut or hazelnut oil for enhanced nutty flavor.
Replace half of the whole eggs with egg yolks.  Cakiness in brownies is partly due to the use of whole eggs.  Egg whites have a drying effect on baked goods.  In converting a brownie recipe I did, I used 2 egg yolks for each whole egg.  
Replace part of the granulated sugar with corn syrup. They hygroscopic nature(ability to absorb moisture and retain it) of the corn syrup will create a more dense moist result.  I think I replaced about half the sugar with corn syrup.  The added moisture of the corn syrup will help to off-set the loss of moisture from removing the egg whites.


Method/Technique:

Heat chocolate, butter, cocoa powder over a hot water bath until chocolate is melted.  Stir to blend to a smooth consistency.  Set aside to cool slightly.

Beat egg yolks/whole egg with sugar and corn syrup until thick, light in color, and mixture forms a "ribbon".  

Combine dry ingredients and make a well in the center.

Temper chocolate into egg/sugar mixture.  Mix to blend well.  Add to dry ingredients and fold together just until moistened.  Pour into greased pan and bake until set.