Alternatively, I’m trying this too, to try and establish the connection between CosmosDB and a LogicApp (the API Connector is already created) in the following way:

const processWorkflow = new azure.logicapps.Workflow('process-url', {
    location: resourceGroup.location,
    resourceGroupName: resourceGroup.name,
    parameters: {
        "$connections": JSON.stringify({
            // "defaultValue": {},
            // "type": "Object"
            "value": {
                "documentdb": {
                    "connectionId": "/subscriptions/639952d4-dcc0-49f4-a588-d59bfb63cf4b/resourceGroups/shortener-rg0620430b/providers/Microsoft.Web/connections/api-connection-to-cosmosdb",
                    "connectionName": "logicapp-comosdb-connection",
                    "id": "/subscriptions/639952d4-dcc0-49f4-a588-d59bfb63cf4b/providers/Microsoft.Web/locations/westeurope/managedApis/documentdb"
                }
            }
        })
    }
});