I think some do if you ask. I know for a fact that the founders of the company I work for would ask for intros to some of the LPs as part of the fundraising process (we do Enterprise b2b sales so they were mostly looking for sales prospects).
Protip: if an investor (potential or otherwise) tells you to let them know if there is something they can do for you, look through LinkedIn and ask for intros to sales prospects
Because their LPs usually insist on confidentially as a condition of investment. Some institutional investors do disclose which VC funds they participate in, though.