Rashed Al Qamzi, who I believe is a supervisor at Al Ain Wildlife Park and Resort (believed to be in Riyadh, Saudi Arabia), says that sand cats do purr but in a slightly different and more subtle way to domestic cats. I will presume that he has first-hand experience of listening to the sand cat purr in captivity. They also meow, he says. This may be learned in captivity as a cat-to-human vocalisation but that is a guess.
However, the books1 that I have do not mention that the sand cat purrs. The best authority on wild cats says that sand cats “growl and spit much like domestic cats, but some of the other vocalisations are quite different from those of domestics”.
The sand cat has a loud bark like that of a small dog. Interesting note: my tabby cat barks like a dog sometimes! He is a domesticated feral cat and I raised him.
The sand cat bark is a sharp call or cry. It can be repeated rapidly. The sand cat bark is believed to be a call over distance seeking a mate. In captivity the barking call is made at close range when cats are introduced to each other.
The sand cat also gurgles which is a friendly close contact call. As stated, there is no mention of the purr from the authors of this fine book on the wild cats. Neither does Wikipedia mention a sand cat purr.
1. Wild Cats of the World by Mel and Fiona Sunquist.