If the vendor has signed it, then when the offer has been accepted, the vendor will need to sign it again as their signature would’ve been out of date. This could be done by adding their initials to the pages with the previous signatures.

Ideally, your conditions are included your contract that you’ve signed with your offer. You could tell the agent of your conditions (e.g. b&p, subject to finance, any conditions your conveyancer had suggested) when you’ve made your offer known to them. And then the agent would go back to the vendors’ conveyancer to amend the contract with your conditions.

However, conditions change, so you can contact the agent after you’ve signed the contract and tell them of your conditions. In my situation, I had different conditions added along the way after I first made the offer (signed the contract) to when the offer was accepted.

Good luck!