You can trigger a runbook. That runbook can run on Hybrid Worker which could be on-premises server. The runbook will run on that environment and log the alert in your on-premises alert tool.
As Log Analytics alerts are logged as data in Log Analytics you can query that information on certain intervals from your on-premises server via PowerShell/C# or other methods. The data is logged in Alert table.
I would recommend the first approach as it is more flexible and you run runbooks only when there are alerts.
Let me know if I understood the scenario and the proposals are what you are looking for.