I've resolved the issue now. There were two things I needed to consider: mikee's answer about needing to await the variable, and needing to parse the variable as JSON before I could use it in code.

The solution being:

let jwtSecret = await pm.vault.get("jwtSecret");
let jwtObject = JSON.parse(jwtSecret);
var signedJWTToken =  KJUR.jws.JWS.sign(null, header, data, jwtSecret);