Can children get a scarlet fever rash without having strep throat?

Yes.

Scarlet fever is a bacterial infection caused by Streptococcus pyogenes, or Group A Streptococcus (GAS). The rash associated with scarlet fever is caused by a toxin produced by the bacteria, and is not directly caused by the strep throat infection itself. Therefore, it is possible for someone to develop a scarlet fever rash without having strep throat, although the rash is usually associated with strep throat infections.