Yes, you can add the Update Payment Method button in the email notifications. Open the email template, click on EDIT MESSAGE and hover over the area where you'd like to add the button. Now click on the (+) icon, choose Add Button >> Update Payment Method and hit Publish

Once this is done, this email when sent to customers will have the Update Payment Method button. On clicking this, customers will be taken to a page where the payment method information can be updated.

Here's a video for your reference. 

If you are using the V1 email notifications, here's the article you can check.