Firstly, according to the documentation, isset() expects a variable. It can't evaluate expressions, as you are trying with isset($request['active_year'] == "1"). Secondly, this is more related to PHP itself than Laravel.
You should split the checking into two parts: first, check if the variable $request['active_year'] exists. You can do that using isset($request['active_year']) or with !!$request['active_year'](using double-not operator). Then, check whether the value is equal to 1 with $request['active_year'] == "1". Note that using == is a loose comparison. If you wish to also check if the variable is of type string, you should use === which is a strict comparison.
Considering the above, I would write:
if(!!$request['active_year'] && $request['active_year'] === "1") {
    $activeYear = 1;
    $currentActiveYear = AcademicYear::where('active_year', 1)->update(['active_year' => 0]);
} else {
    $activeYear = 0;
}