Yes you can easily request statuses for a set of envelopes through the DocuSign API. I recommend going through the DocuSign Developer Center which introduces tools such as the API Explorer and the API Walkthroughs in the Getting Started section.
Take a look at the 5th (middle) walkthrough here, titled Get Status of Envelopes - http://iodocs.docusign.com/apiwalkthroughs
The walkthrough shows you the api call in 6 languages (PHP, Java, Javascript, C#, Python, Objective-C) for getting status of a set of envelopes. Using URL params you can filter by status and date.
The API call you need to make is:
Method
GET
URI
/accounts/{accountId}/envelopes
Optional URL params
- from_date (url encoded)
- status (filter by real-time envelope status)
As mentioned the from_date needs to be URL encoded. For instance, to filter all envelopes that are sent, delivered, or completed since March 1, 2014, you would append the following to the URL:
/envelopes?from_date=3%2F1%2F2014&status=sent,delivered,completed