According to wikipedia, it was bought by Takamatsu and operates as a wholly owned subsidiary, though admittedly I don't think that counts anymore for this title: https://www.kongogumi.co.jp/.
According to the standard established in other cases, that would probably make Takamatsu the oldest company.
From the linked article: “With mounting debts, the company was absorbed into a bigger construction conglomerate in 2006 – but continues to pair traditional temple building techniques with the latest technology.”