test-ethers.js

//link to smart contract https://ethfiddle.com/Q4mg9xHlzR

let address = "0x03b6b93cf23f04a32aa94fe1ef96cae383900ea7";

let abi = [
    {
        "constant": false,
        "inputs": [
            {
                "name": "newMessage",
                "type": "string"
            }
        ],
        "name": "update",
        "outputs": [],
        "payable": false,
        "stateMutability": "nonpayable",
        "type": "function"
    },
    {
        "inputs": [
            {
                "name": "initMessage",
                "type": "string"
            }
        ],
        "payable": false,
        "stateMutability": "nonpayable",
        "type": "constructor"
    },
    {
        "constant": true,
        "inputs": [],
        "name": "message",
        "outputs": [
            {
                "name": "",
                "type": "string"
            }
        ],
        "payable": false,
        "stateMutability": "view",
        "type": "function"
    }
]
provider = ethers.getDefaultProvider('ropsten');
wallet = ethers.Wallet.createRandom();
contract = new ethers.Contract(address, abi, provider);
contractWithSigner = contract.connect(wallet);

await contractWithSigner.update('msgString');

You have to attach the provider to the wallet: wallet = wallet.connect(provider)

I think that should be the only change you need. You can also pass Wallet directly into the Contract constructor and skip the read-only contract if you don’t need it.