No age is not a barrier - the biggest barriers are things like :
- the extent to which you want remuneration (ask for a reasonable wage plus expenses and see what you get). What you get will be the same or less than you ask for usually.
- the number of other approaches firms get. I don't know of any internship opportunities at the moment but your's is a good idea and you are sensible to think about it now rather than when firms have already decided.
- How keen you are/talk about your degree if relevant/why consulting interests you/try and be specific in terms of why you have approached their firm.
- Networking and personal contacts can work quite well at this level. I managed an internship a few years back for a relation of my director for example. This was at a big firm - they were about 20 years old. It can be done.
Writing to firms and making enquiries is another route.
The MCA might be able to help you as well.
You have to sound career minded - let them know you are considering consulting as a long term career path and that you want to get a taste of life in a management consultancy. Make age work for you by being keen, interested, lower cost than senior people etc. etc. It need not be a problem unless you make it one.
You may find a small research project that you can help them out on and so offer them something in return. Ideally you might get a chance to shadow a consultant in a meeting. This offers them little in return, but if you show them respect and they don't see you as a pest, then you would find it more fun.
I did my internship with a law firm and even sat in on a case in the high courts. It was excellent. It made me realise that I would be more suited to a career in consulting. I helped with a research project, and got my travel costs back with a small wage. Whilst I ended up as a Management Consultant it was good to think through the career options ahead of me.
Good luck with your search - this might be a fuller answer than expected.
Do not be put off if people initially ignore you because of your age. Ask around - don't take 'no' responses too seriously - they might be very busy, not busy enough or simply too worried about what other people might think if they give you an internship.
Listen to their reasoning - if it is something you can improve on - then you will know for next time. If is not something you want to change (i.e. your age/sex/racial origin etc.) then quietly ignore them and think about who you can approach next.
