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A raw transaction is just all the transaction parameters encoded in the RLP format. A raw transaction consists of (in this order):

• Nonce;
• Gas price;
• Gas limit;
• To address;
• Value;
• Data.

You don't need the private key of the sender: to get the raw transaction, you can simply encode it using RLP(Nonce, Gas Price, Gas limit, To, Value, Data). Here's an example implementation in JavaScript (using ethereumjs/rlp):

import { encode } from 'rlp';

const transaction = {
  nonce: '0x1e7',
  gasPrice: '0x2e90edd000',
  gasLimit: '0x30d40',
  to: '0xbd064928cdd4fd67fb99917c880e6560978d7ca1',
  value: '0xde0b6b3a7640000',
  data: '0x'
};

// Raw transaction as Buffer
const rawTransaction = encode([transaction.nonce, transaction.gasPrice, transaction.gasLimit, transaction.to, transaction.value, transaction.data]);
console.log(rawTransaction.toString('hex'));

This results in the following raw transaction:

ec8201e7852e90edd00083030d4094bd064928cdd4fd67fb99917c880e6560978d7ca1880de0b6b3a764000080

The only part where a private key is required, is to sign the transaction. A signed transaction is basically the raw transaction with the v, r and s signature (of the raw transaction) added to it: RLP(Nonce, Gas Price, Gas limit, To, Value, Data, v, r, s). The chain ID is encoded in the v parameter (see EIP-155).