No, you cannot file RTI to a private institution.
You can file for RTI to public authorities which come under section 2(h) of the RTI Act 2005:
(h) "public authority" means any authority or body or institution of self-government established or constituted—
(a) by or under the Constitution;
(b) by any other law made by Parliament;
(c) by any other law made by State Legislature;
(d) by notification issued or order made by the appropriate Government, and includes any—
(i) body owned, controlled or substantially financed;
(ii) non-Government organisation substantially financed, directly or indirectly by funds provided by the appropriate Government.