Yes -- the network could soft-fork to require post-quantum transaction signatures. Naively, you could make it so that each transaction would put the hash of it

Yes -- the network could soft-fork to require post-quantum transaction signatures. Naively, you could make it so that each transaction would put the hash of its parties' post-quantum signatures in an OP_RETURN output, and distribute both the transaction and the post-quantum signatures to the network. Miners would package the transactions into blocks, and separately validate them with the associated post-quantum signatures and distribute them as ancillary block data (e.g. the coinbase transaction could contain a Merkle tree root that authenticates all the block's extra post-quantum signature data).