In order to create a signed distribution build, your client would need to provide you with 3 things: the provisioning profile, the distribution certificate, and the private key associated with the distribution certificate. With those 3 files, you would be able to set up your Xcode project for manual code signing. Select the provisioning profile the client provided and then do a distribution build.
Alternatively, you can provide the client an ipa that you have signed and have them resign it themselves, but to make it easier you would want to use a tool like iResign.
Finally, most companies who have a 3rd party do their development will add you as an admin role on their Apple developer account. It would allow you to use Xcode's automatic code signing to create the build. You would simply change the team in your project settings to be the client's account, and Xcode would do the rest.