As far as the privatization of space goes, Space X has been at the forefront. However, there are companies in India as well that have started focusing on that

A Chennai start-up, Agnikul Cosmos, is on its way to build India’s first private small satellite rocket and will seek help from the Indian Space Research Organisation (ISRO) for conducting tests.
After the Indian National Space Promotion and Authorisation Centre (IN-SPACe), an autonomous body under the Department of Space, announced it would help private players gain access to ISRO infrastructure, the IIT Madras incubated start-up received a much required boost. 

The rocket is named ‘Agnibaan’ and according to their website it will be a two-stage LOX/Kerosene vehicle with a third ‘baby stage’. 

Another company in the private space sector in India is Skyroot. According to the website, the company plans to build technologies for ‘responsive, reliable and economic access to space’. Skyroot hopes to make spaceflight as regular, reliable and affordable as air-flight. 

It is founded by former engineers of the rocket design centre of ISRO. ‘Veterans from Indian Space and Defence programs who bring forth over three centuries of collective experience in building rocket systems, are an active part of our journey’, website states. 
The company’s rocket is ‘Vikram’. It is named after Dr Vikram Sarabhai, the father of Indian Space Program. It is a series of launch vehicles especially crafted for the small satellite market. ‘Built on common architecture and covering a wide range of payloads, they offer the most affordable and on demand ride to space’, the website states. 

The website has models of three rockets - Vikram I,II,III. The first one will be capable of carrying a payload of up to 225 kg in orbit up to 500 km. Vikram II will be able to carry 410 kg and Vikram III 580 kg for the same orbit range. 

With brains such as ones from ISRO itself working on making these rockets a success, the day might not be far when India has its own version of Space X!