Vichy is at the middle of France. You can arrive by plane and then go by train to Vichy:
(1) Paris (3h by train) : Roissy Charles de Gaulle Airport or Orly Airport
(2) Lyon (2h by train) : Saint-Exupéry Airport
(3) Clermont Ferrand (30 mn by train): Clermont-Ferrand Auvergne Airport

It is not always staightforwrard to go from the airport to the train station.
From Paris to Vichy, direct trains leave from the train station Paris-Bercy
Here you will find detailed instructions of how to go from Roissy Charles de Gaule Airport to Paris-Bercy train station.
You can Buy your train tickets through the website of the SCNF train French.