Yes it is possible. In fact, the Internet access for management network enables you to access Internet on your OpenStack servers, e.g. controller, object storage, etc. On the other hand, having Internet through provider network enables your instances (i.e. VMs) to access Internet.

Actually you don't need internet access on your management network, only for setup, update, API's access etc. If you like you can have isolated management network and only on net node you must have connection with provide network to give public access to your vm's. Personally i suggest to route internet access on management network because is very helpfully.