Yes, the Department of Homeland Security and its subagency, the U.S. Citizenship and Immigration Services, is in the business of providing genealogy services. They even have a brochure explaining what they do.
Here is a description of the information required for a search:
The USCIS Genealogy program can only fulfill record requests that include a valid file number, such as a naturalization certificate number. Most researchers will receive the necessary file numbers from their USCIS Genealogy Program index search results. However, researchers who have located file numbers independently may use them to make a record copy request.
If your ancestor was born less than 100 years ago you will also need to provide proof of his/her death.
If you do not have a file number, or you are unsure whether or not USCIS maintains a file for your ancestor, please submit an index search request.There is a fee for the service depending on the record format: $20 for an Index Search, and either $20 or $35 for a Record Copy Request.
We have recorded five videos at the Brigham Young University (BYU) Family History Library with instructions about finding your immigrant ancestors, all posted on the BYU Family History Library YouTube Channel. Here is the most recent video:
Understanding Immigration for Genealogists - James Tanner