Thanks for your enquiry.
It would be normal for the Buyer and Seller to sign the Contract themselves.
However, provided their Solicitor has their authority, the Solicitor is entitled to sign the Contract on behalf of their client. This would normally entail the Solicitor phoning the client and asking their authority for him to sign the Contract.
I hope this assists and sets out the legal position.
However, please let me know if you require any further clarification.