The “old law period” lasted from 1790 to 1906. The Naturalization Act of 1790 permitted free white people of “good moral character” to apply for naturalization. Only a handful of changes to those requirements occurred during the next century or so, most notably the passage of the 14th Amendment. However, and to directly answer your question, there were never any regulations or federal enforcement around the Naturalization Act and its amendments. Federal district courts were conferred jurisdiction to grant naturalization however there was no uniform process about that. The type of evidence required, the petition submitted, and even the eventual “certificate” varied from court to court. Super chaotic, super prone to fraud.

That all changed in 1906 with the establishment of the Federal Naturalization Service. They’re the ones who set up the first regulations around form, evidence, access to duplicate records, etc. That agency would eventually become the INS, and then its contemporary incarnation the US Citizenship and Immigration Service. The USCIS has a great write up about all this on their website.