How do I automatically split an audio file where a tone occurs?

This is actually possible using Audacity's Beat Finder analysis plug-in when combined with a bandpass filter.

1. Duplicate your mono audio track.
2. Find the tone's exact frequency.
   1. Select the tone you want Audacity to detect and choose Analyze → Plot Spectrum.
   2. Set the size to 16384.
   3. Move the mouse pointer over the highest peak.
   4. Look for the "peak" frequency.
3. Use a bandpass filter to remove everything but the tone.
   1. Select the entire tone detection track and choose Effects → Nyquist Prompt.
   2. Type (reson s 1000 1 1) into the box.
   3. Replace 1000 with the frequency you got from Step 2.
   4. Click OK.
4. Optionally, apply the Amplify effect, setting New Peak Amplitude to zero.
5. Use Analyze → Beat Finder. Start with Threshold Percentage equal to 100; reduce that percentage until Audacity successfully detects the tone.
6. Use File → Export Multiple to actually save the marked segments as separate files.