Yes. Each claim is a wallet transaction, so you pay the normal network fee for the chain. Keep a small balance of the chain's coin in your wallet.
On Solana every claim also creates a receipt that needs a deposit of about 0.0013 SOL. That is why Solana prizes have a minimum of 0.005 SOL. A smaller amount is not paid on its own: it rolls over or adds up until it is worth claiming.