As is often the case, Raw Story gets more wrong than right.
There is no "law", obscure or otherwise, that gives the House the right to see everything they want in the Mueller investigation.
There are a couple of court decisions relating to the disclosure of grand jury testimony where impeachment is in play. It should be noted that these cases arguably could be distinguished on their facts by a judge who was so inclined. For example, in the Watergate decision, the court stated that "We think it of significance that the President of the United States, who is described by all parties as the focus of the report and who presumably would have the greatest interest in its disposition, has interposed no objection to the District Court's action [allowing the disclosure to Congress of a report prepared by the grand jury]." Also note that the grand jury in that instance prepared a report and asked the judge to transmit it to Congress -- something that hasn't happened with respect to the Mueller investigation.
There is an argument that under the US Code (not obscure and oft discussed here on DU and elsewhere) there are circumstances under which grand jury testimony can be disclosed "at a time, in a manner, and subject to any other conditions that it directs—of a grand-jury matter: (i) preliminarily to or in connection with a judicial proceeding." The argument would be that impeachment is a judicial proceeding and even before it is commenced, this exception would apply under the "preliminarily to" provision.
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