Generally, yes, it is possible to employ professional genealogists to carry out research on your behalf. A lot of people do it, especially when they hit 'brick walls' in their research or need copies of records held in archives that they cannot access, and many - perhaps most - are very happy with the results they get.
You should be aware that employing professional genealogists can be expensive. For this reason, it is important to ensure that you provide them with all the information that you have (together with copies of any documents etc.) right at the start of the process. Otherwise you might find that you are paying them to find out what you already know!
My advice would be to have the research carried out in agreed stages, and to always agree a quote for the expected cost of each stage in advance!
The Society of Genealogists have produced a guide with hints and tips for Employing a Professional Genealogist. While written with researchers in the UK in mind, many of these tips are applicable more widely.
Obviously, you are going to want to check that the person you will be employing to carry out your research is competent before you hand over any money. Many countries have certifying bodies and/or professional bodies that represent professional genealogists and researchers. For example,:
In the UK
In the USA (and also internationally)
In Australia & New Zealand
Unfortunately, as far as I can see, there doesn't appear to be an equivalent body in India, and (at the moment) India isn't one of the countries of residence listed on the search page of the APG website.