Excuse me, does the lawyer's contract have legal effect if it is sent by mail?

This is of course legally binding. The contract does not have to be in black and white with clear physical attributes, and the electronic version can also be used.

The essence of a contract is that both parties agree on the meaning. Even if there is no written text, the contract is established and has legal effect. What's more, you already have a contract that he emailed you. If there is a dispute, you only need to prove that he sent you this email, and you really agree to reply to his contract. You can ask him to perform the corresponding duties according to the contract.