Hello,
I have personally not been, so this is information from the internet (in Japanese) and I cannot guarantee it is 100% accurate. Please take it with a grain of salt as the information may be old.
Matsushima-Kaigan
It seems that Matsushima Kaigan Station only has stairs! So, you may need a taxi… However, a popular option is to go to Shiohama station and take a ferry through the islands. The company has at least three accessible (“barrier free”) boats. You can contact them via their website: https://www.marubun-kisen.com/english/matsushima/list.html
Hon-Shigama
According to this list from the Miyagi Government and this personal blog, it appears to be accessible:
Tazawako
Is accessible. Elevators and an Accessible toilet:
http://jikoku.toretabi.jp/map/pdf/t_tazawako.pdf
Kakunodate
Is accessible. Elevators and an Accessible toilet:
http://jikoku.toretabi.jp/map/pdf/t_kakunodate.pdf
Aomori (not Shin-aomori)
Only the east entrance is accessible. There are no listed elevators, but the (Japanese) JR site mentions that there are freight elevators they use, so you will need help.
Atsumi-onsen
There are three platforms, one for a train that pulls in next to the ticket gates which will be accessible, and then a central platform for two other lines. To get to the central platform you must take a tunnel from the ticket gates under two of the train lines and then stairs up to the central platform. This platform (and trains) are not accessible.
You may be able to use the one that stops next to the ticket gates by going one station further and then changing so you ae at the right platform, but that is impossible to tell in advance as it depends on how willing the staff are that day.
Sakata
According to JR (Japanese site) it has elevators and an accessible route to the train.
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If you are interested in renting a car, please tell is the general area and what type of wheelchair your wife uses as there many types of accessible cars – some only can take a manual chair but not a power wheelchair, etc.