Does India make cryogenic engines?

Yes, India does make cryogenic engines. An example is the CE-20 cryogenic engines that are used in GSLV. You can read about the GSLV Mk III – A Comprehensive Overview of Heavy Lift Launch Vehicle in the given link.

Further readings:

1. Indian Space Research Organisation [ISRO Notes for UPSC]
2. Mars Orbiter Mission (MOM) – A Complete Overview of Mangalyaan Mission