According a plausible scenario, genetic code originated in the RNA World, meaning that aaRSs would have originated in protocells. It is not likely to be any earlier because aaRSs are part of the translation machinery, and that as a relatively complex system could not have arisen in "simpler" pro-protocells. These papers are interesting:
Article Horizontal transfer of code fragments between protocells can...
Article Origin and Evolution of the Universal Genetic Code