What type of Logic App are you using?
Standard Logic apps can run in Azure ARC enabled servers or App Service Environment allowing for integrated support for virtual networks and private endpoints. I have run logic apps in fully isolated environments, similar to what you are trying to achieve.
https://learn.microsoft.com/en-us/azure/logic-apps/logic-apps-overview#resource-environment-differences
P.S. Prior to the standard logic apps, the Consumption mode could run in integrated service environments, which achieved similar results, though much more costly/