Interesting question, but I don't think so. The VLSC is not an attorney appearing for either party but has (according to your question) only provided some casual information to one party who wants to act as his/her own attorney and referred the other party to a lawyer (if I understand your question).
So, I don't see any conflict of interest, especially if the goal of seeing this conflict is pro se Party "A" trying to get represented Party "B"'s lawyer disqualified. I don't see this VLSC has having an attorney-client relationship with either party, and presumably the VLSC informed the pro se lawyer that it was giving them general resource information, not agreeing to act as his counsel (like Avvo responders).
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