The Modi government’s approach to vaccines is based on the central pillar of RSS ideology: the task of the state is only to help big capital. Anything else including planning is “socialism”. In the case of vaccines, it means no attempt to get the companies—both public and private sector together—to plan what was required for a quick vaccination program; put in the money and the necessary supply chain. Instead, it believed that India’s private pharmaceutical industry would do all of this on its own.
It’s not just about the production capacity of India. India currently has two vaccines - Covishield (Oxford & AstraZeneca developed, manufactured by Serum Institute of India) and Covaxin (Indian Council of Medical Research & Bharat BioTech, manufactured by Bharat BioTech). Sputnik V is a new addition from Russia.
While it would be wise to blame the country for the under production of Covaxin, because it’s a home grown vaccine and public money was involved in its development. It could’ve been manufactured by more than one manufacturer in the country.
But, in case of Covishield, only SII has the license to manufacture it. If the IP rules are to be waived, it could be manufactured by others in the country.
The whole point is, countries like India have the capability to mass produce vaccines to satisfy domestic needs and to help other countries that don’t have the necessary facilities to do so. The rich countries have secured enough vaccines (How rich countries are making the pandemic last longer) and in the case of UK, they have clause in the vaccine contracts to give it more preference, should the producer face shortages (link).