Can a web ordering app send push notifications on iPhone?

Yes, since iOS and iPadOS 16.4. It only works for web apps the customer has added to the Home Screen, and only after they tap a button to allow alerts.

• The customer must add your site to the Home Screen first.
• The site can ask for permission only after the customer taps something, like a subscribe button.
• Many customers will not add the site, so do not rely on push alerts alone.
• Plan another way to tell customers about their order.