There are classes in the Java API that support communicating with an
HTTP server.  They are java.net.URL and java.net.HttpURLConnection.  See
the javadocs for details.  Basically, you create a URL, call its
openConnection method, and cast the result to an HttpURLConnection do do
do HTTP-specific stuff.

There are limits to the capabilities of these standard API classes.  If
you need more functionality, a fuller interface, or any of a few dozen
other things that HttpURLConnection doesn't provide, then the Jakarta
Commons HttpClient package will probably do what you want.  Get it from
jakarta.apache.org and see the documentation on that site.  (And, of
course, come back if you have specific problems.)

That leaves the question of whether HTTP alone can do what you want
anyway.  Theoretically, it can.  HTTP's GET method will retrieve a file,
and its PUT method will upload a file.  That said, though, the number of
HTTP servers that actually implement the PUT method is negligible, and
doing things this way is extremely unlikely to work.  It's more likely
that you need to figure out exactly how you're expected to upload files,
and let us know that information before we'll ave any clue how you might
upload files to the web site.  You've said it's not FTP, and it's very
unlikely to be plain HTTP PUT requests.  Is it FrontPage extensions?  
Something else?