Are there any moving parts to a scramjet engine?

No, the scramjet engine does not have any moving parts.You can read about the Hypersonic Technology Demonstrator Vehicle (HSTDV) – unmanned scramjet technology of DRDO in the given link.

Further readings:

1. Advanced Technology Vehicle – India, one of the 4 Countries To Develop Scramjet Engine
2. Reusable Launch Vehicle – Technology Demonstrator (RLV – TD)