Can You Get Grunt Funeral Skull / Halo Avatar Armor Without Xbox Live?

No. Both items require Xbox Live:

• Grunt Funeral Skull (Halo 3): Originally unlocked via Xbox Live Marketplace (free download) or found in-game on Legendary (mythic skull). The download method needs Live; the in-game method does not, but the skull itself is tied to the profile.
• Halo Avatar Armor: Avatar items are purchased/downloaded via Xbox Live Marketplace—requires an active connection and (historically) Gold subscription.

Workarounds (Limited)

• Offline Profile + Legendary Playthrough: Earn the Grunt Funeral Skull in-game on Legendary without Live (Halo 3 only).
• Modded Console (RGH/JTAG): Can inject unlocked items offline—violates ToS, risks ban, not recommended.

For legitimate access, an Xbox Live (now Xbox Network) connection is mandatory for marketplace content.