"She can plan less activity" and "she can plan fewer activities" are equally correct and almost mean the same thing. "Activity" is a word that is often used both as a "countable" noun and an "uncountable" noun. That is why both "she is planning an activity" and "she is planning some activity" sound good.
"Less activities", however, is something you should not say. Since "activities" is plural, it cannot be uncountable. It must be countable. You can have "two", "three", "more", "few", or "fewer" activities, but not "less". "Less" can only be used with uncountable things. The reason for this is usage, not grammar. We just don't use "less" that way even though we do use its opposite, "more", that way.