I was approached to build an iOS app for a company. What would be the best way to go about this when developing and managing the project?
Yes!! Client needs to create their own Apple Developer Account for current and future applications.
They needs to have paid account to publish App on store. Meanwhile either they can add you as developer or share the original credentials with your so you can sign the application with development and distribution certificates for Notification and publish.
It's a one time JOB, but trust me best option for You and your Client as well.