Maybe try local politics, state legislatures, state agencies. Internships that work on writing, regardless of whether they're strictly law-related would be good.

That said, I wouldn't worry about doing something specifically law related. There are lots of areas of law, and lawyers usually end up specializing in something. So, think about what topics you are interested in and care about. Someone who wants to be an environmental lawyer, for example, would benefit from having an environmental internship (even if it isn't law-related) and it would make them look more interesting on law school applications. There are lawyers in the music industry, there are patent lawyers, medical malpractice lawyers; lawyers who specialize in plants, and the sea, and construction, and space, etc. Plenty of people get into law school and are successful despite (or because of) having engaged in things that did not have "law" on them.