Lockdown mode hides App Store on iPhone

In no way are you ever allowed to access the App Store through a Web Browser. Lockdown Mode disables many services and there are quite a bit more that you will not be able to do when it is enabled. It is not designed for normal operation of your phone and is only intended for extreme protection for very few people who are targets of attacks for who they are or what they do. Think of a high level official that would be a target of a mercenary attack from a foreign adversary. If that is you, then downloading anything is going to be severely restricted and also includes attachments that you would get in a message.

Lockdown Mode is an optional, extreme protection that’s designed for the very few individuals who, because of who they are or what they do, might be personally targeted by some of the most sophisticated digital threats. Most people are never targeted by attacks of this nature.

About Lockdown Mode - Apple Support

You probably have it disabled in Screen Time. Make sure Installing Apps is set to Allow at Settings > Screen Time > Content & Privacy Restrictions > iTunes & App Store Purchases > Installing Apps. That is the most common reason why the App Store would appear to be missing.