You can no longer purchase Fallout 3 DLC digitally on Xbox 360. The Xbox 360 Marketplace closed in July 2020, removing the ability to buy or re-download add-ons like Operation: Anchorage, The Pitt, Broken Steel, Point Lookout, and Mothership Zeta.
Bethesda released a "Game of the Year Edition" containing all DLC on disc. Used copies are available on eBay, Amazon, and retro game stores. This is the most reliable method.
Individual DLC discs were also released (rare). Verify the disc is for Xbox 360 (not PC/PS3) and check for scratches.
If a friend owns the GOTY Edition or DLC discs, you can install the content to your hard drive from their disc. However, the license stays tied to the disc/profile, so you may need the disc inserted to play.
Avoid "digital code" listings for Xbox 360 DLC—they are almost certainly scams or revoked codes. Stick to physical media or alternative platforms.