I found a similar card for my ancestors. It was tied to the "Declaration of intention to naturalize". The National Archives can find the actual petition. I live in N. California and was able to contact a local branch of the national archives to obtain the document. Apparently these records can be held at local branches. The Declaration of Intention documents will give you information about your relative, but won't tell you whether or not they completed the process. The U.S. Citizenship and Immigration Services provides the search to determine if they completed the naturalization process.