Does Stanford University offer internships for Indian students?
They do not have any specific summer internship program for Indian students.
During my stay at IITB, the only person that I have ever heard who interned at Stanford was a senior of mine in Electrical Engineering Department- Raaz Dwivedi. He worked on load balancing in computer networks. Mind you, he was IR (Institute Rank) 1, was awarded the President's Gold Medal and had a passing out GPA of 9.95/10. Moreover, from what I had heard, he had got his internship because of good relations with professors from EE department at IITB.
Other than that, Stanford usually doesn't take IIT students for internship because of logistic issues. A professor at Stanford was willing to take another senior of mine for summer internship but due to mis-match of summer quarter at Stanford and summer vacation at IITB, the offer could not be extended. The professor wanted the student to work in his group from late June to September (almost 3 months), but at IITB, our Autumn Semester starts from mid July. So, it wasn't possible.
Moreover, you can go out of your way and apply for programs such as SN Bose Scholarship provided you are the topper of your department, and a professor at Stanford is willing to take you for the stipulated duration, then I believe it would be smooth for you.