My answer is wrong!
I just realized that this transmission of Eisenhower was reflected off of a balloon in 1960, but the question is really asking about the transmission in 1958 from SCORE
1958-006A. See this answer.
Here is the wrong answer, left for its potentially interesting content:
Sure you can! Eisenhower's recorded transmission appears within the first minute of the YouTube video below. This seems to be part of the "Recently rediscovered audio recordings from the JPL archives..."
Recently rediscovered audio recordings from the JPL archives, for instance, highlight the lab's involvement in America's early attempts at satellite communications.
One of these vintage recordings comes from NASA's first communications satellite, Project Echo, which bounced radio signals off a 10-story-high, aluminum-coated balloon orbiting Earth in 1960. This form of "passive" satellite communication -- which they dubbed a "satelloon" -- was an idea conceived by an engineer from NASA's Langley Research Center, Hampton, Virginia, and a project managed by NASA's Goddard Space Flight Center in Greenbelt, Maryland. JPL's role involved sending and receiving signals through two of its 85-foot-diameter antennas at the West Coast Goldstone tracking station in the Mojave Desert.
The Echo transmissions include a greeting from President Dwight D. Eisenhower, explaining how Echo fit into the U.S. program of peaceful space research accessible to other countries, and a message from then-Senator Lyndon B. Johnson imagining a "not too distant future when one man, one program, can be seen and heard simultaneously in every living room of the world.
This video is one of several in a group of interesting vintage space items in the NASA Jet Propulsion Laboratory YouTube channel: