Are Naot (cork sole) or other Leather strapped sandals permitted on Yom Kippur. It seems that Rav Obadiah permits it although I am unclear of the specifics and other opinions.
There seems to be a machlokes regarding shoes with leather straps (nothing else made of leather). This a quote from a QA in shulchanaruchharav.com
May one wear shoes that have only leather straps? Some Poskim rule it is forbidden to wear it. However other Poskim rule it is permitted.
The sources of the Machlokes are
 Maharam Shick 316; Mateh Efraim 614/2; Divrei Shem Chadash p. 36
 Maharikash in Erech Lechem 614; Peri Chadash 614; Peri Hadama 1/43; Zechor Leavraham 9; Moed Lekol Chaiy 10/40; Birkeiy Yosef 614; Yifei Laleiv 2/6; Kaf Hachaim 554/75; 614/10; Gesher Hachaim 21/4-1; Pnei Baruch 15/1
Regardin Naot, I believe that those usually also have an inner leather sole which would not be in the scope of the aforementioned QA and probably part of the general prohibition of wearing leather shoes. Some Poskim rule it is forbidden to wear it. However other Poskim rule it is permitted.