To download to disk, you can use download.file().
If you want to include an image in a Notebook, you can use it as a link:
---
title: "R Notebook"
output: html_notebook
---
Example image:

end
If you knit this as rmarkdown, you will get an html document with your avatar. If you open the source of this html document, you will notice that the image was fully included in your document (so if it's a big image, you will get a big result).
I expect it's possible to only include a link to an image in your document, and not download it to your computer, but I don't know of an easy way (haven't really looked though).