Yes, it is possible depending on the SDK you are using. For instance, in pysui, if a 'sponsor' isn't identified, the transaction is paid by the sender, and gas is used from the sender's Sui account. Alternatively, you can provide a 'sponsor', and the gas will be taken from their account; however, the 'sponsor' would need to sign the transaction.

Sui transactions are public, meaning the transaction details are visible to others. To avoid tracking, you might consider solutions like Zero-Knowledge Proofs that can offer some anonymity, though they are complex and may not be directly applicable to all scenarios.