The Court's official website provides transcripts of oral arguments on the same day an argument is heard by the Court. Same-day transcripts are considered official but subject to final review. Transcripts are available for the cases argued during the 1968 Term and subsequent Court Terms.
Oral argument transcripts in paper or microform are available at the Law Library of Congress. See the library's finding aid for specifics on their collection and how to request transcripts, e.g., by US Reports citation, Term, and/or Docket Number.