Can Azure Container Instances be invoked using triggers?

You can trigger the container to run either from Logic Apps, or Azure Function. There are samples for both cases:

Event driven using function: https://github.com/Azure-Samples/aci-event-driven-worker-queue

Trigger using Logic Apps: https://github.com/Azure-Samples/aci-logicapps-integration

Another way I've found to do this:

Use an Event Grid subscription to trigger an Azure Automation runbook (by means of a webhook created on that runbook). In my case, the Event Grid reacts to a storage account event, namely blob creation. The runbook in turn starts the container group by means of a Powershell script like the following:

Select-AzureRmSubscription -SubscriptionName "yourSubscription"

Invoke-AzureRmResourceAction -ResourceGroupName yourResourceGroup -ResourceName yourContainerGroupName -Action Start -ResourceType Microsoft.ContainerInstance/containerGroups -Force