There are three ways to match a user's email address with a LinkedIn profile.
Have a special partnership with LinkedIn which allows you access to non-public API functions. This is very hard, and per my understanding, they only consider big guys (Startups with serious funding, established companies...)
Retrieve the user's details from another social network via email (Facebook used to allow that, not sure these days), and then perform a faceted search in LinkedIn's API using details other than the email (name, job title, location...) . You will get accurate results lots of times.
Use a 3rd party service such as fullcontact.com in order to get you those results.
I'm not sure if you'll be breaking any LinkedIn TOS, it all depends on what you're trying to achieve. So make sure you thoroughly read their TOS.