Definitely put your resume in and, if possible, contact the Executive Director of the firm - be open with your lack of experience but also convey your interest in the field.
Having provided technology services to some of the largest law firms in the world (I build document assembly/automation systems), I love the environment. However, understand that part of what paterson says above is true.
Not so much that attorneys are "dumb" - dumb, in my opinion, being a gauge of intellectual capacity - but they are typically busy and bill by the hour. Therefore, technology is often perceived (and may be) and hindrance to what they do. Should you get the job, remember that. If you can learn to navigate the partners at a law firm, you can probably deal with the most difficult personalities you come across.
Usually, they are very intelligent and also very demanding.
Once again, I love the environment but I know some people who really struggled when faced with a strong-willed and angry attorney. For me, I understood their frustration, didn’t really concern myself if they were abrasive, did my best to solve the problem and not “inconvenience” them.
Fair/Not Fair, Right/Not Right, and “he shouldn’t talk to me like that,” are all terms that are meaningless in that type of environment.
But if/when you win over some of your most demanding and least patient clients, the benefits can be nice.
Matthew Moran (career blog and podcast below)
Career Advice with Attitude for the IT Pro