Does the industry still use SABR to calibrate a cap/floor spot vol surface under the current low rate environment?

Not sure what you call spot vol surface... Cap/floor isn't a very liquid product. There is some liquidity though. It does trade in the broker market. I know some guys who work wedge trades like there is no tomorrow... In my understanding some dealers don't even trade this stuff any longer. I am mostly familiar with USD. I know there is some activity in EUR also.Sure you can use something like a SABR model or maybe a shifted variant thereof or a better model if you have one. It's not entirely clear how you would calibrate this. It's more likely that you calibrate it in relationship to the STIR option market in the front and then relative to the swaption market elsewhere somehow. It's pretty heuristic to be honest.