Can I purchase cellular data for my Apple Watch?

Only if your Apple Watch is a GPS AND Cellular model. In addition your cellular provider must support the Apple Watch. Below is a link to the list of current providers.

Apple Watch - Carriers - Apple

Does your Apple Watch have a Red Ring ⭕️ or a Red Dot 🔴 on the Digital Crown?

If it does, then you have a cellular model, and you can ask your iPhone cellular provider to add the Apple Watch to your iPhone contract (for a monthly fee).

No Red Ring or Dot, then no cellular radio chip in the Apple Watch.