Doppler-effect can be visible in Audacity’s spectrogram view if you increase the resolution, (aka “window size”), from the default setting, see … Doppler Effect for High School Physics - #3 by Trebor
However the object will have to be moving at a speed which is around 1/20th the speed of sound, (~15m/s), to be measurable. The Doppler-effect at say walking-speed, (~1.5m/s), will probably be too small to measure.