Where do I get a Scoreboard Code?

It comes from your meet's timing operator or team — usually posted with the meet program or shared in a team group chat. SwimBoard doesn't generate or assign codes itself.

Does SwimBoard replace the official scoreboard?

No. It's a live companion view of the same Event/Heat data shown on the pool deck's scoreboard, meant for following along from the stands. Official results still come from the meet's timing system.

Why does it ask for Lock Screen permission?

Only if you choose to start a Live Activity. It's entirely optional — you can follow the meet from inside the app without ever enabling it.