Is there a way to track email client within Marketing Cloud?

There is an 'Email Performance by Device' report available, but you will need to have Discover enabled on your account (which is included in corporate and enterprise editions). However, this provides limited analytics and you can only filter by date (not send).

The best method would be to add Litmus tracking code to your emails, which enables you to see a breakdown by client, and also how much time they spent reading your emails. Refer to screenshot below. You can sign up for a free trial at litmus.com.

Device and client reporting is rather limited in SFMC, which is a shame. Aside from Analytics Builder reports which off limited onsite, you can get Device/Client reporting and deeper insight using Inbox Tools (a.k.a ReturnPath)

You could alternatively use Google Analytics to at least get you device insight from the Click activity of URLs