How to do transaction on a dapp without any wallet

Essentially the user will have to sign each transaction with the owned private key. One option is to use the sendRawTransaction() method from web3js: https://github.com/ethereum/wiki/wiki/JavaScript-API#web3ethsendrawtransaction

When you say without a wallet what do you mean? A Dapp will use Web3, most popular is the web3.js library. It lets you make JSON-RPC calls to your node.

To get started: https://github.com/ethereum/web3.js/

It gives you a wide range of APIs that are very easy to use. Refer to this document to know which APIs you can use: https://github.com/ethereum/wiki/wiki/JavaScript-API

There are other libraries as well use as the web3.j for Java, Nethereum for C# web3.py for Python. Implementing on web3.js has a lot more community support and tutorials that you can follow.

You can even get started on smart contracts that would run on your nodes to enable you to make a wide range of Dapps. For that, check out Solidity.

You should use Web3. Then you can do access a node and do transactions from your own website. You can also use web3 for python.