Is there way to custom program an F key?
I would use Spark to program the F15 key to do what you want. It is free (although donations are accepted). The website for Spark says "Mac OS 10.9" but I find it works with 10.11.4 here. 

Here is what Spark looks like on my Mac. You create a new hotkey from the File menu. In your case, you'll be making a new "Keyboard" type of hotkey. I am including a picture that shows what the new key would look like if your email address were the same as mine. You get to pick your own hotkey-- I picked Control-F12 because I don't have an F15 on my keyboard.

There are plenty of commercial hotkey apps that do more than Spark but "free" is nice and your needs will be met with it. See the link at the end of this answer.

Note: I am not affiliated with Spark or its developers in any way. I have used the product on multiple machines for many years with great results so I am speaking first-hand about how it works. 





Link to shadowlab.org, where you can download Spark