Not the VSS. If it was, the speedo wouldn't work as well.
This is a
very common problem on the 1996-1998 Fords (not just Mustangs). The worm gear in the odometer did not get enough lube from the factory. It dries up and strips the tiny plastic gear inside the odometer.
There are
MANY shops that specialize in Ford Odometer/speedometer repair. It should cost about $100 as a bench job. Likely $150-175 taking the car in.
For an example of a company that has done mine:
United Speedometer and Electrical > Home
I am sure an online search will yeild others (local and mail in).
If you ever think about selling the car, get the Odometer fixed before it can become an issue with any new possible buyer.