The correct usage is both your examples is "any problems" (plural) because it is uncertain how many problems there could be. If I tell you "you can choose any book you like from this pile", it is certain that you only get to keep one book. If I tell you "you can choose any books you like from this pile", the number is uncertain. It could be one or ten. If you want to use singular, it would more commonly be "you won't have a problem getting rice"

as a native english speaker, I find myself tending to use "any problems" more than "any problem", but yes both are correct and there is no real reason for using one over the other

Hi Jane! Both are generally correct. Most people will use either one interchangeably. 兩個都可以用。