Zao Kokusai Hotel
蔵王国際ホテル
The hotel answers the question twice on its own site and the answer for the shared baths is no. Its FAQ and the remarks line on its hot spring page both say tattooed guests are asked to stay out of the large bath and the open-air bath. What it does have is three reservable private baths, booked at the front desk after check-in and open 07:00 to 21:45, and those are the whole reason this house is listed. Book one, or two, and skip Hachiemon-no-yu.
タトゥー・刺青がある方は、大浴場・露天風呂のご利用をご遠慮いただいております。
Best for: Three different private tubs in one building, including a barrier-free one you can reach by wheelchair
Private baths: Yama-no-megumi-yu, a cypress tub for up to 7 at 4,400 yen for 45 minutes; Mori-no-megumi-yu, granite, up to 3, 2,200 yen; Sato-no-megumi-yu, step-free from corridor to tub with the tub raised for wheelchair users, up to 6, 3,300 yen. Booking is first come, first served after check-in, and the hotel takes 1,100 yen off if you reserve through its own site. All three come with a toilet, a dryer and toiletries. Day visitors can book them too, from the day before by phone, but the hotel charges the private bath on top of the day bath entry of 1,320 yen for adults, taken between 12:00 and 15:00, so a tattooed day visitor pays for a shared bath the same page tells them not to use. The hotel is part of the Zao-no-Kaze group with Zao Shiki no Hotel and Omiya Ryokan, which publish the same refusal.