Audacity does not have it's own codec at least not in the past. LAME is what Audacity pointed one to. Still needs to be installed and you still need to point Audacity to where that codec actually lives on your machine.

In fact here's the text from Audacity on the preferences specific to where LAME is located: ---------------------------
Export MP3
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Audacity does not export MP3 files directly, but instead uses the
freely available LAME library to handle MP3 file encoding. You must
obtain lame_enc.dll separately, by downloading the LAME MP3 encoder,and then locate this file for Audacity. You only need to do this once.

Would you like to locate lame_enc.dll now?
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Yes No
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-Scott