It depends on your university, its professors and the institute policies.
There is significant research happening in universities, but in India, it's most prevalent in the top-ranked institutes/universities, which are mostly government institutes. Can you get involved in initial years? Yes and no. It's difficult to answer without knowing if your uni is decent w.r.t to the world or just the country. I study at IIT Delhi (Indian Institute of Technology, Delhi) and I'll tell you my experience:
Yes, you can ask professors to get involved in research in your initial undergrad years. You'll most probably not have the required background, but there are professors willing to help you learn and develop and/or explore interest(s). You'll not contribute significantly but it'll help you start learning. To reach to the level of actually publishing something, it'll happen later in undergrad. A freshman is just getting started will college and sophomore with their major. It is after that that you'll start taking advanced courses - but who's stopping anyone from learning!
For summer and winters, many students apply for research opportunities in other institutes and universities for research interns in summer. A lot of people from the top IITs and/or IISc also go to research interns abroad. At IITD, we also have institute programmes to do projects in summer. A lot of information about such programmes is shared over mailing lists or informed by your seniors in college. Most of the research opportunities are obtained by emailing. Even for research programmes, a better path is emailing someone from the research team first - unless the guidelines state otherwise.
If you're not in IIT/IISc/IISER, you can mail professors from these institutes to get involved with them. However again, that requires you to show more dedication since there are already students from that institute itself showing interest. Very few to none have dedicated research programmes for external students.
Most professors will deny, some will be open to short projects while some might ask you to dedicate longer times. Usually the shorter projects do not provide a new learning curve, so research projects are usually the latter ones. Landing an opportunity depends on how you approach professors and show interest. They do not need you, you need them; show interest, be willing to learn.
In conclusion,
Can you have research experience in your undergrad in India? Yes.
In initial years? Depends on you and under whom you apply.
Are researches necessarily done by college professors or there are other groups of researchers who provide opportunities out of school? Not all researchers are professors but most of such opportunities are available to grad students so your best bet is first reaching out to professors in or outside your university since they're more open to working with students.
