Can You Buy Stuff on Xbox Without Xbox Live? Digital No, Physical Yes
Digital purchases: No. You need an Xbox Live account (free) to access the Xbox Store, browse, and buy. For online multiplayer, you need Xbox Live Gold (paid).
What Works Offline
Physical discs: Buy at retail, play offline (no subscription needed).
DLC codes: Redeem codes bought at retail — but the DLC itself may require an internet connection to download/install.
Summary
Xbox Live (free) required for any digital store access.
Xbox Live Gold required for online multiplayer.
Physical games + retail DLC codes work offline (mostly).