I know that most nitrogen is fixed through industrial processes and bacterial symbiotic relationships. However, are there any plants that can fix their own atmospheric nitrogen?
There are no known plants that fix their own nitrogen. However, there soon may be!
Because N is the major limiting factor in agricultural productivity, there is huge interest in plant systems which can fix their own. It's important enough that the Bill & Melinda Gates foundation have started a project aimed at reducing dependence on fertilisers by giving plants the ability to fix their own nitrogen. There are several potential strategies, the most obvious being encouraging more species to form symbioses with nitrogen fixing bacteria.
However, there's another proposal which builds on recent discoveries about the mechanism of nitrogen fixing in bacteria: we now know how the crucial enzyme complex is made (Rubio & Ludden, 2008). As a result, there are many people calling for efforts to engineer the system directly into a plant organelle (e.g. Beatty & Good, 2011; Godfray et al., 2010).
So, in 10-15 years time you can check back and the answer to this question might have changed! The most likely answer then will be "just the usual suspects: rice, maize, wheat".
As far as I know, all biotic nitrogen fixation is performed by prokaryotic organisms such as Rhizobium. I don't know of any plants which can carry out this function on their own.
Plants can't use atmospheric N2 because it is held essentially inert by the nitrogen triple bond. The process of reducing N2 to NH3 which is usable by plants can be summarized:
N2 + 8e- + 8 H+ + 16 ATP -> 2 NH3 + H2 + 16 ADP + 16 Pi
(where Pi is a phosphate group)
Nitrogenase catalyses the reaction reducing N2 to NH3 by adding H+ and electrons. The whole process requires 8 ATP and is therefore energy intense.
In order to perform this coversion, bacteria require sufficent carbohydrates from decaying matter or plant vascular tissues (this is how Rhizobium derives energy from the host plant).
However, I should add that bacteria often have a mutualistic relation with the plant to perform this function, so in this sense you could say that plants can fix their own nitrogen.
There are also "free living" ammonifying bacteria in soils.
Science and the Garden, eds. Ingram, D.S., Gregory, P.J., Blackwell, 2008