您正在检索str.length()的值之前您已将用户的输入读入str。此时str仍然是空的，因此length变量始终为0，因此循环永远不会运行。
您需要在将输入读入str后**读取str.length()值：

#include <iostream>
#include <string> 
using namespace std;

int main() {
    string str;
    int length = 0;
    char ch;
    char search;
    int count = 0;

    // length = str.length(); // <-- too early here!

    cin >> search;
    cin >> str;

    length = str.length(); // <-- move it here instead!

    for (int i = 0; i < length; i++) {

        //current character
        ch = str[i];
        //cout << ch << " ";
        if (ch == search) {
            count += 1;
        }
    }
    cout << count << " " << search << "'s" << endl;

    return 0;
}

Online Demo
话虽如此，您可以通过使用迭代器而不是索引来消除阅读length的需要：

#include <iostream>
#include <string> 
using namespace std;

int main() {
    string str;
    char ch;
    char search;
    int count = 0;

    cin >> search;
    cin >> str;

    for (auto iter = str.begin(); iter != str.end(); ++iter) {

        //current character
        ch = *iter;
        //cout << ch << " ";
        if (ch == search) {
            count += 1;
        }
    }
    cout << count << " " << search << "'s" << endl;

    return 0;
}

Online Demo
或者，使用range-for loop（内部使用迭代器）：

#include <iostream>
#include <string> 
using namespace std;

int main() {
    string str;
    char search;
    int count = 0;

    cin >> search;
    cin >> str;

    for (char ch : str) {
        //cout << ch << " ";
        if (ch == search) {
            count += 1;
        }
    }
    cout << count << " " << search << "'s" << endl;

    return 0;
}

Online Demo
或者，使用标准的std::count()算法而不是手动循环：

#include <iostream>
#include <string>
#include <algorithm>
using namespace std;

int main() {
    string str;
    char search;
    int found;

    cin >> search;
    cin >> str;

    found = count(str.begin(), str.end(), search);

    cout << found << " " << search << "'s" << endl;

    return 0;
}

Online Demo