From a formal perspective you can certainly use semiconducting electrodes. The question is however, what relevance it has. Unless the electrode is conducting, there will be no injection and hence no current.
In some situations involving semiconducting materials, like gate stacks etc, it's more realistic to have a metallic injection part, then a semiconducting buffer layer, and then perhaps the scattering region.
Another type of system, studied in experiments, might have (heavily) doped semiconductor regions as electrodes, but then again we have conducting electrodes. Still, at the "end" they are probably connected to metal electrodes anyway (connection to external circuitry, for instance).
So, if we are interested in the pure properties of a semiconducting scattering region, it's best to use as transparent and broad band injection as possible, i.e. some metal like gold or even just lithium.
However, thinking about it, perhaps your question is rather, what happens when the bias overcomes the band gap of the semiconductor. That is, if the system is so aligned that the valence band of one electrode overlaps the conduction band of the other. In this case, as Nordland writes, in principle all works out fine - except for the trouble with the band gap in DFT, of course...