Do Japanese people call their money "yen" even though there is no YE sound in their language?
Japanese don't pronounce 円 as "Yen" like "i-en." We pronounce it clearly "えん," though I don't know how to describe Japanese "え" sound by using alphabet and phonetic sybols.
As a side story, I have a memory of having read an episode in some book that 伊藤博文 - Ito Hirobumi, the Japan's first Prime Minister proposed to use the denomination of 円 for Japan's currencies, when the cabinet members of the new government were discussing how to call the unit of new currency that replace old currencies (両, 文, 朱) issued under the Tokugawa regime at the cabinet meeting soon after the launch of the Meiji government. He showed the round shape of a ring with his thumb and index finger, and said "円 is a familiar shape of coins and it signifies PEACE and perfection, which is a suitable name for the new currency."
Provided this is the provenance of 円, it's no wonder to pronounce it as えん - en, not yen.