Organizations that offer online wallets aren't obligated to handle transactions using the actual Bitcoin blockchain; they may instead choose to keep their own off-chain record of transactions between users. While faster and cheaper than on-chain transactions, this system is often less secure, especially since the transaction records are centralized.
Only when a user transfers Bitcoins between an external wallet would an organization utilize the blockchain; this is when a transfer would cost some network fee and take some time to process.