Do most law schools want you to attend their undergraduate school?

They don’t mind but they prefer you to.

Some law schools used to display the represented undergraduate institutions of j.d. candidates and number of students admitted from each on their website. Not now though.
A handful of students from their undergraduate pool will usually be represented in an entering class but since most of the top law schools seek the broadest representation of education, experience, and quality, going to Columbia College will not necessarily make entry into their law school any easier, although being a stellar student at a college like Columbia would. The only real advantage I see is that some Ivy League universities have programs where selected undergraduates from their college are able to work conjunctively and eventually be admitted to their respective law school. In such cases, it is hardly difficult to discern a distinct advantage existing, however, those students are usually the best students anyway.

One caveat–if you attend a smaller, lesser known state college for undergraduate studies, the top law schools in the nation will usually admit one to two students from such colleges whereas you might see anywhere from three to twelve students admitted to Harvard Law from Harvard College, fifteen admitted from Princeton, and one to five from Berkeley in a given year. Meaning, although there is no set pattern for a successful entry, there is something to say about where you receive your education.

The main point here is that you are a competitive applicant, which means high LSAT, high GPA, and rigorous curriculum/studies.

Hope this helps.