This website can pull up historical flight records from 1995 and onward:
Aircraft Tail Number Lookup
It doesn't look like American served LAX-LAS for most of 1999 (or it didn't show up), so I chose December 19, 1999.
N567AM, an MD-83, was one of them. "NR51AA and "NR79AA" are showing up but I can't find what those aircraft were.