Viterbi Decoder - Decoding Continuously Encoded Stream Partially
Since the decoder that you are working with is a black box, yes, adding prefixes and postfixes to the stream to "window-ize" it is appropriate.  The pre and postfixes need to be the actual data from the previous and post windows, and they need to be at least as long as the traceback depth.  If you do this and get rid of the result bits that correspond to the prefixes and postfixes, then you should do fine.

In a normal situation you would not need to add any postfixes or prefixes.  There are only two things you need to do to handle continuous streams of Viterbi-encoded data.

1) occasionally check the state metrics and lower them when all of them get "high".  This works because the only thing that matters is the difference between the state metrics, not their absolute value.  An easy way to do this is to check if all of them are equal to or above half the maximum state metric value.  If they are, subtract half the maximum value from all of them.

2) You have to do tracebacks while you are decoding.  This is do-able in both FPGA's and microprocessors (it's actually quite easy in microprocessors).  Every $N$ transitions (pick an $N$ that works for your architecture) you do a traceback and output $N$ bits.  You have to go back through $M$ bits ($M$ being equal to the traceback depth) before you get to the $N$ bits that you will output.