AWS returns "Invalid Signature" when trying to get Orders

At a guess, it might be that you need to use execute-api instead of Execute-api since the service name will be case sensitive.

Other than that, there are various things that could be wrong with the signature - this is probably a good place to start: https://docs.aws.amazon.com/general/latest/gr/signature-v4-troubleshooting.html

Deriving the signing key is just one step of the process, and any other issues are more likely to be with your algorithm to create the canonical request and string to sign, which will need to match the request parameters exactly. More information is available here: https://docs.aws.amazon.com/general/latest/gr/sigv4_signing.html