I am making an application in C# unity and trying to integrate polkadot wallet in the application. I want to know whether it's possible to use the polkadot wallet browser extension to sign personal transactions.
The maintained project for substrate-based chains and Unity integration is covered here in this substrate seminar and the linked GitHub.
Build .NET and Unity Parachain apps | Substrate Seminar https://youtu.be/IAdvCbK0kQI
Substrate C# Toolchain that generates a NET API and more https://github.com/ajuna-network/Ajuna.SDK
Basic NET API for Substrate (follow-up project of SubstrateNetApi) https://github.com/ajuna-network/Ajuna.NetApi
I hope that helps with the Unity integration.
What you want is a library written in C# like: https://github.com/usetech-llc/polkadot_api_dotnet
However this project does not seem to be well maintained at the time of writing this, so probably you will need to reach out to the contributors or fork it if you want to use it in a modern Substrate application.
I'm under the impression that the browser extension wallet is not meant to be used from outside the browser. So, unless you are using unity for creating a WASM module that can only be consumed from the browser, and then you add some layer so that the Web-Worker that will be serving the WASM module somehow has access to the global scope where the browser wallet is injected, then unless that's the case... I'm afraid that nope, you can't consume the browser wallet from a unity project.
However, have you considered integrating your unity application with Parity Signer instead?