I just tried it on my Kindle Keyboard 3G, using the built-in browser, and when within WiFi range.
On the Gutenberg web site, it seems like it is working. And it does put a .mobi file in the Documents folder of my Kindle. But this mobi file is NOT the book from Project Gutenberg, but rather a text file that can't be opened from the Kindle. When I connected my Kindle to my PC, and opened the .mobi file, in the Kindle documents folder, with Notepad, here's what was in it:
<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML 2.0//EN">
<html><head>
<title>403 Forbidden</title>
</head><body>
<h1>Forbidden</h1>
<p>You don't have permission to access /cache/epub/2701/pg2701.mobi
on this server.</p>
<hr>
<address>Apache Server at
www.gutenberg.org Port 80</address>
</body></html>
I tried this for two books.
When you don't have a general purpose computer (like one running Windows, Linux, Mac OS, etc.), the bad news is that the Kindle Keyboard works best when you can partner it with a computer. However, a friend can download books for you and then email them to your @kindle.com address. That's works well, although you first have to go into Manage My Kindle on the Amazon web site and allow the friend's email address access to yours. Doing that maneuver from the Kindle Keyboard (as opposed to a PC) is probably possible, but difficult.