There are instructions for inserting Emoji's into the subject line of your email but not directly into the text of the message . I was able to do that following the instructions of one of the helpful Constantcontact support reps by doing the following:
Search for emoji in emojipedia.org > copy the emoji and paste it in the campaign. It will automatically adjust to be the correct size.
Using emoji's from other sources did not work.
Thank you for reaching out to the Constant Contact Community. It's my pleasure to assist you today.
Thank you for providing the feedback regarding emoji's. I have submitted your feedback to our team. Please feel free to reach out should you need anything else. Thank you for choosing Constant Contact. Have a great day!