Can google vision API accept external image URL?

Yes, you can do this with any image as long as it is smaller than 4Mb. It does not have to be on Google Cloud Storage.

Here is an example using the Golang client library:

// Copyright 2016 Google Inc. All rights reserved.
// Use of this source code is governed by the Apache 2.0
// license that can be found in the LICENSE file.

// [START vision_quickstart]
// Sample vision-quickstart uses the Google Cloud Vision API to label an image.
package main

import (
    "fmt"
    "log"

    // Imports the Google Cloud Vision API client package.
    vision "cloud.google.com/go/vision/apiv1"
    "golang.org/x/net/context"
)

func main() {
    ctx := context.Background()

    // Creates a client.
    client, err := vision.NewImageAnnotatorClient(ctx)
    if err != nil {
        log.Fatalf("Failed to create client: %v", err)
    }

    image := vision.NewImageFromURI("https://www.denaligrizzlybear.com/assets/images/scenic-denali.jpg")

    labels, err := client.DetectLabels(ctx, image, nil, 10)
    if err != nil {
        log.Fatalf("Failed to detect labels: %v", err)
    }

    fmt.Println("Labels:")
    for _, label := range labels {
        fmt.Println(label.Description)
    }
}

Here is the function on the Godoc:https://godoc.org/cloud.google.com/go/vision/apiv1#NewImageFromURI

The docs state:

NewImageFromURI returns an image that refers to an object in Google Cloud Storage (when the uri is of the form "gs://BUCKET/OBJECT") or at a public URL.

Answer for python users.

def detect_labels_uri(uri):
    """Detects labels in the file located in Google Cloud Storage or on the
    Web."""
    from google.cloud import vision
    client = vision.ImageAnnotatorClient()
    image = vision.types.Image()
    image.source.image_uri = uri

    response = client.label_detection(image=image)
    labels = response.label_annotations
    print('Labels:')

    for label in labels:
        print(label.description)
# [END vision_label_detection_gcs]