If it needs to be framed, if fireblocking needs to be installed, electric outlets added to meet the code, lights, etc you need a permit.

Is there a permit on record for the finished portion of the basement?

Yes there was a permit for finished basement, when he had bought this was confirmed. It already had been framed with high quality wood, when we purchased it. Its just not drywalled. Is there a way I could find the orginal permit online?

Contact the town where you live they will have a recoed of permits. If all you need is elecrical you can probably hire an electricia to pull the permiy for that and once the electrical is checked you can drywall and where I live no permit required for dywall.

It's a good idea to pull a permit. As mentioned above, they will spot any issues like fireblocking, electrical, etc.