My daughter received vaccination in a private clinic near Jaidah Roundabout. The price of the vaccination is not covered by insurance, and you will have to pay for it (it varies from vaccination to vaccination).
The Private Clinics do not store vaccines, they have to order for them, so you might have to go and get an appointment to come a certain day of the week.
Try your luck, or call beforehand to be on the safe side and ask the cost as well. It will save you a trip.

For telephone number and names, google will help.