Yes, citizens from the Vatican City may apply online for their Indian E-Visa here: IndiaVisaApplication.com
Its done in 3 easy steps:
1) You complete a pre-application form (less than 5 minutes)
2) We send you a link with a form that you can complete in less than 10 minutes.
You also need to send a passport photo and a copy of your passport in PDF format (our visa experts will give you guidance to make the process as simple as possible).
3) Once we have your details online, the passport photo and the passport photo information, we will process your eVisa.
The whole process shouldn’t take more than 48 hours.