Here is a list of these Government Centers

Vaccination centres For Yellow Fever For International Travellers

Delhi:
Health Organization, Palam Airport. (Tue. and Thursday 1400 - 1600 hrs) Tel. 3295507.

Public Health Laboratory, Municipal Corporation, Town Hall, Alipur Road, Mon & Fri 1000 - 1200 hrs. Tel. 252 2058

Dr. Ram Manohar Lolia Hospital (Wed & Sat 1100 to 1300 hrs) Tel: 3365525 / 3735337

International Inoculation Centre, Mandir Marg, New Delhi - 110001

Mumbai:

Health Office, Mumbai Airport.

Port Health Organisation, Seamens Medical Examination Organization, Nav Bhawan

Calcutta:

All India Institute of Hygiene and Public Health.

Health Organization, Dum Dum Airport.

Port Health Organization.

Cochin: Port Health Organization

Goa: Urban Health Organization, Panaji.

Hyderabad: Institute of Preventive Medicine

Jamnagar: Irwin Hospital

Kandla: Port Health Organization.

Kasauli: Central Research Institute

Lucknow: Balrampur Hospital

Chennai: King Institute of Preventive Medicine, Guindy.

Port Health Officer / Assistant Port Health Officer / Assistant Airport Health Officer, Port Health Organization.

Marmugao: Port Health Organization

Porbandar: Office of the Civil Surgeon

Pune: Department of Preventive & Social Medicine, BJ Medical College

Vishakhapatamam: Port Heath Organization.

Reference
http://www.indiaatbest.com/india_welcome.htm

Unfortunately, due to bureaucratic red tape and governmental apathy and indifference, most of these centers have restricted working hours, need for prior appointment and rude staff whose attitude leaves a lot to be desired.
Hence, many enterprising private doctors have been providing this vaccination to people needing the yellow fever vaccine. This vaccine is called STAMARIL (manufactured by Sanofi Aventis at France) and imported under license of Drug Controller Authority of India.
When provided along with the WHO International Certificate of Vaccination, the vaccination is internationally accepted with a validity of 10 years.