I would guess the answer to be yes - I believe they run 365 days per year - it just might not be all 4 ferries (but that could be any given day anyway )
I would HIGHLY recommend a private water taxi on this day. We arrived on carnival day last year - no way we could have gotten in and gotten cars if it were not for our fantastic on-island villa reps. You will not be able to get into town at all - there's a great little market now - I totally forget the name- but it was by far our favorite little place all week!
Some agencies on STJ will still be open - the only down side is luggage but if you have carry on only and don't mind walking through tons of people it is still doable
I would not chance it if it were me. Town is closed to through traffic in the morning through the mid afternoon. The barges will probably be filled with lines because of heavy flow from STT.
If I had to do this, I would arrange for a car to be left where I could get to it in or near town and forgo renting on STT, and use the people ferry.