You can use the message button on a member's profile to send them an email. You'll need to add them into your network, but you can add a message at the time you send the request.
Pro users get 5 personalised messages to non contacts per month & 10 standard connection requests per day.
Basic members get no personalised messages but 5 standard connection requests per day.
Once they've accepted your invite you'll be able to message them any time.