Crear un proyecto de tipo blank:

ionic start PruebaCamara blank

Acceder a la carpeta del proyecto

cd PruebaCamara

Instalar el plugin de la cámara:

ionic cordova plugin add cordova-plugin-camera
npm install @ionic-native/camera

home.page.html

<ion-header>
  <ion-toolbar>
    <ion-title>
      Prueba de la cámara
    </ion-title>
  </ion-toolbar>
</ion-header>

<ion-content padding>
  <ion-button (click)="hacerFoto()">Hacer foto</ion-button>
  <img [src]="foto" *ngIf="foto" />
</ion-content>

home.page.ts

import { Component } from '@angular/core';
import { Camera, CameraOptions } from '@ionic-native/camera/ngx';

@Component({
  selector: 'app-home',
  templateUrl: 'home.page.html',
  styleUrls: ['home.page.scss'],
})
export class HomePage {

  foto: any;

  constructor(private camera: Camera) { }

  hacerFoto() {
    const options: CameraOptions = {
      destinationType: this.camera.DestinationType.DATA_URL
    }
    this.camera.getPicture(options).then((imageData) => {
      this.foto = 'data:image/jpeg;base64,' + imageData;
    }, (err) => {
      console.log(err);
    });
  }

}

app.module.ts

import { NgModule } from '@angular/core';
import { BrowserModule } from '@angular/platform-browser';
import { RouteReuseStrategy } from '@angular/router';

import { IonicModule, IonicRouteStrategy } from '@ionic/angular';
import { SplashScreen } from '@ionic-native/splash-screen/ngx';
import { StatusBar } from '@ionic-native/status-bar/ngx';

import { AppComponent } from './app.component';
import { AppRoutingModule } from './app-routing.module';

import { Camera } from '@ionic-native/camera/ngx';

@NgModule({
  declarations: [AppComponent],
  entryComponents: [],
  imports: [BrowserModule, IonicModule.forRoot(), AppRoutingModule],
  providers: [
    StatusBar,
    SplashScreen,
    Camera,
    { provide: RouteReuseStrategy, useClass: IonicRouteStrategy }
  ],
  bootstrap: [AppComponent]
})
export class AppModule {}

Acceso al proyecto en GitHub

https://github.com/javiergarciaescobedo/EjemploCamaraIonic