I think it depends on the country. There is a person at my school who got a grant to Indonesia and most of hers was done online. For me, I got a letter saying they accepted me, that my grant is contingent upon completing the medial form and sending in the official grant acceptance form that I should get sometime soon from Argentina, and a lot of stuff about how Fulbright was so great. :wink: There was also the medical form, instructions for completing it, and a reminder to send transcripts to IIE when I graduate. I think that was about it. This was for an ETA so it may be different for research grants and between different countries.
For now, I am supposed to wait for more information from Argentina. That was about a month ago... but my grant doesn't start until March 2010 so I am not really worried about it. I'm sure if your grant starts in September they will be much faster.