I guess with App ID you mean something like "com.company.product".
Getting an Application for OS X to the App store is not an easy task (compared to iPhone/iPad).
To sign the App you need the public/private Key pair and a couple of certificates from your client.
The convenient way to upload the app to apple is done using Xcode.
Does the client use Xcode? If your client uses Xcode, then your client can do all the signing stuff. Then your client needs just the source code.
If your client needs a signed binary, then it's not obvious HOW he gets the app to the Appstore. There is a tool which can be used to upload the binary: "Application Loader.app"
(Here is a similar SO question which describes the toolchain: How to submit an iOS app WITHOUT XCode?)
If you need to deliver your results to your client as a signed bundle then you need all Certificates from the client. Your code must have all entitlements set. Uploading this code without testing your entitlements on your local machine is like driving a car blindfolded.
However: If you need to deliver a signed binary to your client, then you need all certificates.
If you will distribute the code to the client, it's not a problem delivering an unsigned binary to the client. Apps can be executed without code signing. Even without an valid AppID your code may be executed.
If your client has knowledge about Mac development, this should be no problem. If your client't doesn't know anything about Mac development, you should get access to his Mac-Developer account and do it for him.
Conclusion: The AppID is just a string to identify the app. If your client does the code signing stuff and uploading to Apple by himself using Xcode, then you need nothing else.
If you should use iCloud or App-Sandboxing (Entitlements) then you NEED certificates from your client.