I'm assuming you meant to say "some internship experience but no full-time work experience".
Few things - there is always a possibility, but it's very unlikely you will be considered as part of the class for the recruiting season that you just missed, since those classes are likely filled. Here is your best option
1) Get some full-time work experience, ideally at a top company. It doesn't have to be consulting, but a brand name is important.
2) Your best odds are if you get a recommendation from a current consultant. If you know someone who is currently at a firm, have them recommend you and that's probably your best bet to get an interview.
As for the process - I've seen people submit a regular application, but make sure that you get a recommendation from a current consultant around the same time. Otherwise, your application will likely end up in the reject pile, given the number of applications they receive. Maybe do 1-2 conversations with current consultants before that as well, to have a few more people know you and think you are a nice person. I'd recommend having your contact connect you directly to the recruiting manager.
Now, once you get the interview, it's all about how you do on the case.