You don't need work experience to get into uni for law. Lots of people on my course didn't have any. I would try the high street firms as opposed to big ones, they'll be more willing. Also in your letters try saying you want to "shadow" a solicitor as opposed to "work experience", this makes it seem like you are happy to sit quietly as they go about their day (and that they don't have to give you anything to do...).
While I did have extensive work experience, as others have said, it shouldn't take up too much of your PS anyway. Law is not a vocational course, it's taught as an academic subject and not everyone who reads Law will go on to be a lawyer. The LPC is the vocational training part of becoming a solicitor, NOT the law degree, and you need to show that you understand this distinction in your PS. There's nothing wrong with saying you intend to be a solicitor but you should focus on why you want to study Law as opposed to practice it.