Before commencing a Legal Practice Course (LPC), you must register with the Solicitors Regulation Authority (SRA).
The SRA requires all applicant to disclose all convictions and all cautions, even if they'd otherwise be regarded as 'spent':
http://www.sra.org.uk/students/student-enrolme nt.page
I very much doubt that a caution for shoplifting, at the age of 13, would affect your chances of successful registration with the SRA, but you can read their criteria here:
http://www.sra.org.uk/documents/students/stude nt-enrolment/characterguide.pdf
(In particular,
read Section 6.3).
The Bar Council (for those who wish to become barristers, rather than solicitors) will apply similar criteria.
There are plenty of career paths for law graduates outside the direct 'administration of justice', where you'd not be required to declare your caution. See here:
http://www.prospects.ac.uk/cms/ShowPage/Home_p age/Options_with_your_subject/Your_degree_in_l aw/Job_options/p!empFbib
Chris