No, MBBS is not mandatory to become a radiologist. You can go for the same through B.Sc as well as you can pursue diploma courses, bachelors course, masters course and certification courses in the field of radiology. The courses offered in radiology are mentioned below:

Certificate Courses:
Certificate in Radiography
Certificate in Radiology Assistant
Certificate in Radiography Diagnostic

Diploma Courses  2 years
Diploma in Radiography and Radiotherapy
Diploma in Radio-diagnostic Technology

Bachelors Courses  3 years
B.Sc in Radiography
B.Sc (Hons.) in Medical Radiotherapy Technology
Masters Courses  2 years

Post Graduate Diploma in Radiotherapy Technology
Post Graduate Diploma in Radio-diagnosis and Imaging Sciences
Post Graduate Diploma in X-ray Radiography and Ultra-sonography


Hope this helps!!!

Good Luck!!!