Inner tubes are dirt cheap so I'd always go with a new one after a puncture. I carry a couple of spares so any tyre fitter can get me going again if I have a flat.
As for sourcing them in the right size, speak to King's Lynn Tyres -
They have all sizes in stock as they're the go-to place for stock-car competitors who race at the Saddlebow track (and run with tubes fitted because they help keep the tyre on the wheel in shunts). I needed one replacing after a puncture and it was about £7.50 for one the right size for a 175/80/14 tyre.
As for driving at 100mph+ on tubed tyres, unless you're on run-flats you're in trouble if a tyre blows out at those speeds, tube or no tube.