The NASA JPL video Corporal: America’s First Guided Ballistic Missile shows that it used an Acid/Analine engine.
That would have been both challenging and unpleasant to reproduce. So instead, the video below Re-Launching the First Science Rocket to Space says after
02:10 that an "M motor" was used.
Question: What is an "M motor?"
It looks like Estes Model rocket engines only go up to G: https://en.wikipedia.org/wiki/Model_rocket#Performance (from this answer)
A NASA scientist, physics professor, and model rocket hobbyist recreates an historic rocket launched by aeronautics pioneer Frank Malina. The first WAC Corporal dummy round was launched on September 16, 1945 from White Sands Missile Range near Las Cruces, New Mexico. After a White Sands V-2 rocket had reached 69 miles on May 10, a White Sands WAC Corporal reached 80 km (49 mi) on May 22, 1946 — the first U.S.-designed rocket to reach the edge of space (under the U.S. definition of space at the time).
Screen shot from the YouTube video Re-Launching the First Science Rocket to Space