You are not allowed to use your westlaw or lexis account over the summer for firm-related research. Read the terms; you can DEFINITELY get flagged and be charged personally for the searches.
If you're allowed...
Do you know of any schools with contracts that allow for students to use Lexis and Westlaw for private firm work? I know that some allow you to use it for pro bono/non-profit work, law review during the summer, RA jobs for professors and other academic use (moot court/mock trial) or extracurricular work, but I have never heard of a contract that would allow a student to use their school access for summer employment with a private firm that would otherwise have to be paying for the searches.
In this market, I don't think OP has a great shot at a paid firm gig in South Carolina and should certainly make back-up plans for unpaid jobs.