Do what you would do once you start your MBA - reach out to people who work in consulting firms that you are interested in, ask them to meet for coffee, enquire what it takes to get into their firm and mention that you are keen to do an internship pre-MBA. If they are smart, they will like your pro-activeness and find you an opportunity in their firm. If there is no opportunity, you have now built a rapport that can at least give you a head start once you do start your MBA.
Don't get bogged down by formal processes in consulting firms, there is almost always a way around these so called formalities. What is the worst that will happen - the person you are meeting will tell you they cannot help you. Try someone else.
Boutique firms are of course better because of lesser formalities, but the important thing is to try in larger and smaller firms nevertheless.
Don't forget - this is what you should ideally start doing since first day in MBA anyway if you are keen on consulting, you might as well start the process early.
Hope that helps.