By default, you can use the ` (backtick key) to open it up,
but if you have other kinds of keyboards and you think it could be a pain to do it this way,
keep reading.

---

Open up Input.ini inside

Then insert these 2 lines:
In place of "<Whatever_keybind>",
type any key you want to press to open up the console (e.g., F10).

With the console open in-game, you can:

• Start recording a replay:
  
  demorec "Replay_Name"
• Pause the recording:
  
  demopause
• Stop the recording:
  
  demostop
• Play the local replay:
  
  demoplay "Replay_Name"

and here's where to find them on your PC:

This guide can be also used in other Unreal Engine games that support this feature,
not only Insurgency Sandstorm.

❶ExtremelyStiffت for my initial find:
https://gtm.steamproxy.vip/app/581320/discussions/0/6086066536281862501#c6086066536284284342

From the official unreal docs:
https://docs.unrealengine.com/5.1/en-US/using-the-replay-system-in-unreal-engine/