As with most media capture capabilities available on Android, we can use an Intent to leverage the the built-in application. To use the built-in camera application, we create an Intent like this:

// Path to where we want the file and what to call it
String imageFilePath = android.os.Environment.getExternalStorageDirectory().getAbsolutePath() + "/myfavoritepicture.jpg";
// Create a File object out of that
File imageFile = new File(imageFilePath);
// Create a Uri out of that
android.net.Uri imageFileUri = android.net.Uri.fromFile(imageFile);

// Create the Intent that triggers the camera
android.content.Intent i = new android.content.Intent(android.provider.MediaStore.ACTION_IMAGE_CAPTURE);
// Tell the camera application where we want the resulting image saved
i.putExtra(android.provider.MediaStore.EXTRA_OUTPUT, imageFileUri);
// Start the Camera
startActivityForResult(i, CAMERA_RESULT);

To get the resulting image, we need to implement an onActivityResult method in our sketch:

PImage cameraImage;

protected void onActivityResult(int requestCode, int resultCode, android.content.Intent intent) { 
  super.onActivityResult(requestCode, resultCode, intent);
  if (resultCode == RESULT_OK) {
    // We know the location via the imageFilePath String so load it into a standard Processing PImage
    cameraImage = loadImage(imageFilePath);
  }

Here is a full example:

final int CAMERA_RESULT = 0;

String imageFilePath;
File imageFile;
android.net.Uri imageFileUri;

PImage cameraImage;

void setup() {
  imageFilePath = android.os.Environment.getExternalStorageDirectory().getAbsolutePath() + "/myfavoritepicture.jpg";
  imageFile = new File(imageFilePath);
  imageFileUri = android.net.Uri.fromFile(imageFile);
}

void draw() {
  if (cameraImage != null) {
    image(cameraImage,0,0,width,height); 
  }
}

void mousePressed() {
  android.content.Intent i = new android.content.Intent(android.provider.MediaStore.ACTION_IMAGE_CAPTURE);
  i.putExtra(android.provider.MediaStore.EXTRA_OUTPUT, imageFileUri);
  startActivityForResult(i, CAMERA_RESULT);
}

protected void onActivityResult(int requestCode, int resultCode, android.content.Intent intent) { 
  super.onActivityResult(requestCode, resultCode, intent);
  if (resultCode == RESULT_OK) {
    cameraImage = loadImage(imageFilePath);
  }
}