WASHINGTON—House Democrats, poised to reactivate a probe of President Trump’s campaign ties to Russia, said this week that an investigatory priority is learning more about a Republican activist who sought 2016 Democratic candidate Hillary Clinton’s private emails, and whether he served as a conduit between the Trump campaign and Russia-tied groups.

“That is an area of continuing interest for us,” Rep. Adam Schiff (D., Calif.), who is expected to become chairman of the House Intelligence Committee when Democrats regain control...