When male immigrants were naturalized, their minor children also automatically received citizenship. Between 1855 and 1922, their wives did, too.
If your online and microfilm searches are fruitless, check archives at all three levels. The records may still be with the court where your ancestor filed, or they may have been sent to a state archives or other repository. Some archives and other official stewards of naturalization records have posted indexes and documents online. A Google search could turn these up, and NaturalizationRecords.com is also helpful.
US Citizenship and Immigration Services has copies of all post-1906 naturalization records. To request those, use the agency’s online Genealogy Program.
Naturalization records can be a gateway to finding your ancestor’s passenger arrival list, as they often tell port and date of immigration (though the earlier the records, generally the less detail they contain). In fact, that’s how I confirmed the family story of my great-grandfather Henry Essel’s 1888 arrival through Philadelphia, enabling me to locate the ship manifest recording him and his family.