Not according to anything I’ve read. Many upper-class men in the early republic did wear wigs, but that was a mark of class and not a badge of office limited to law-talkin’ dudes. Here’s a famous painting of John Jay, the first Chief Justice of the United States (I’ve seen the original myself in the Supreme Court building). He is wigless: https://upload.wikimedia.org/wikipedia/commons/7/72/John_Jay_(Gilbert_Stuart_portrait).jpg
Apparently when Monty Python was suing ABC in New York, they played a number of the sketches that had been censored in violation of their contract. With the Poofy Judges’ “waggle my wig” line, the court reporter was laughing so hard he could not record the proceedings.