Registers are the vents that control air flow into a room from heating and cooling systems.

https://www.heatair.org/

A louvered damper enables control of the air, which can be directed in at least one direction or, in some cases, up to as many as four, or diffused altogether. These are usually the best floor registers for air flow in residential spaces.

Registers can be used in walls and ceilings so cold air can fall into a room, and in floors so heated air can rise.

If they become blocked, the air temperature in the room can be greatly affected.

if you cannot find the contractor who can do both then hire two different ones and get the job done.

Wood, cast iron, metal? Style?