If I take a screenshot in-game, is there any way to download it to my computer? In Halo Reach or 4, I can just upload it to my file share and download it from here, but there’s no way of viewing your Halo 3 file share here, or at least a way that I know of. How does one view their Halo 3 file share on their web browser today?
I have the same problem.
Halo 3 fileshares are no longer available online to view or download from. It was said by both Bungie and 343, before and after the franchise changed over that Halo 3’s fileshares were not going to be part of Halo Waypoint or carried on in any way by 343.
At this point the only way to get screenshots/videos from Halo 3 is by using a capture device such as a HD PVR or Elgato Game Capture.
343 cares about as much as H3 as they did making Halo 4 play good. 
Either use a capture card or a program on the PC to extract it. They exist, but I’m not sure if I can link them (Hint: Bing).
The only Halo 3 file shares that exist now are the ones on Bungie.net, but they haven’t been updated since March 31st 2011. They are still the same now as they were back then and cannot be changed.
> Either use a capture card or a <mark>program on the PC to extract it. They exist, but I’m not sure if I can link them (Hint: Bing).</mark>
You are correct, you cannot name the programs used to extract screenshots from the console. Not only is naming them against the Halo Waypoint forum rules, but this is classed a form of modding/content manipulation, which is against not only Halo games’ terms of use, but breaches the XBL user agreement and Terms of Service policies (which OP can lead to being permanently banned from Xbox Live on both your gamertag AND console if found out).