Artic,
you can the common flu vaccine (not Swine Flu Vaccine as it's not available yet) from any Health Center around Doha; pediatric health center for kids or any private hospital.
I do my kids' vaccination at Al Ahli Hospital and this is where they had it last year. I had mine there too.

However, I am reluctant in doing the flu vaccination this year as my kids kept being sick all year long. So, I'm sceptic about its effectiveness...
It's best to discuss with your doctor/pediatrician and do whatever you feel is best for you and your family.

Best