I am building a microbial fuel cell. Can I do CV for the full cell without a reference electrode? Do I always need a reference electrode for CV? Or can CV be done using the anode as working electrode and cathode as both the counter and reference electrode?
You can't control the potential without reference electrode. So, it is important to use reference electrode to get the reproducible CV.
In fact, there are numerous types of electrochemical systems where reference electrode is difficult to have (for example - solid batteries).
Generally, if the potential of your counter electrode is stable and is not changing despite the passing current, you can use it as reference electrode. For example, using microelectrodes, as mentioned above, since the current passing is very low and risk of changing RE potential is weak.
Without reference electrode you can run a slow scan rate CV (typically 1 mV/s for a biofuel cell, depending on the size/surface area of your electrodes) from your OCV to 0 V (short circuit) with the configuration you proposed (with reference connection of your potentiostat connected to your counter electrode). Assuming you reach a pseudo-steady state at each point of your CV you will obtain a polarization curve of your electrochemical cell (Voltage = f(current)) and can then obtain the power curve (P = f(j))you are probably looking for. Of course it does not allow to discriminate between the contribution of your anode and cathode. For the latter you need a reference electrode.
For running CVs you can use a two-electrode or a three-electrode potentiostatic system. In this of two electrodes, you have the working electrode and the other electrode, that acts simultaneously as reference and counter (then current then passes through this and the working (circuit between both) and maybe the potential is not as stable as required).
In the case of the three electrode system, current passes through the counter and the working (the reference is out of this system), and the potential is more easily maintained. As commented before, in the case of ultramicroelectrodes the current that passes is very low (proportional to the area of the working electrode) and if you use a two-electrode system, the current that passes through the ref/counter electrode is very low and probably potential is not affected.
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