Basically you have already everything.
You want to push_back a line into a std::vector if a certain condition is met. The condition is that we can find a key string in the line.
This we can do with a simple if statement:
if(line.find(key) != std::string::npos)
And, the new function would would like the below (I added some comments):
void readFileVector(std::string& fileName, std::vector <std::string>& fileVector)
{
    // Open input file
    std::ifstream file(fileName);
    // Check, if it could be opened.
    if (!file) {
        //if not, then show error message
        std::cerr << "File could not be opened: " << fileName << std::endl;
    }
    else {
        // File could be opened. Now we want to read line by line
        std::string line{};
        // Read the complete file
        while (getline(file, line)) {
            // Search for the key. If we find it, then we can add the line to the resulting vector
            if(line.find(key) != std::string::npos)
                fileVector.push_back(line);
        }
    }
}
But, my guess is that you want to split the lines into parts and then do some comparisons.
For splitting a text lines into single parts, also called tokens, there are many possible solutions. I will show you 4 examples:
#include <iostream>
#include <fstream>
#include <sstream>
#include <string>
#include <regex>
#include <algorithm>
#include <iterator>
#include <cstring>
#include <forward_list>
#include <deque>
using Container = std::vector<std::string>;
std::regex delimiter{ "," };
int main() {
    // Some function to print the contents of an STL container
    auto print = [](const auto& container) -> void { std::copy(container.begin(), container.end(),
        std::ostream_iterator<std::decay<decltype(*container.begin())>::type>(std::cout, " ")); std::cout << '\n'; };
    // Example 1:   Handcrafted -------------------------------------------------------------------------
    {
        // Our string that we want to split
        std::string stringToSplit{ "aaa,bbb,ccc,ddd" };
        Container c{};
        // Search for comma, then take the part and add to the result
        for (size_t i{ 0U }, startpos{ 0U }; i <= stringToSplit.size(); ++i) {
            // So, if there is a comma or the end of the string
            if ((stringToSplit[i] == ',') || (i == (stringToSplit.size()))) {
                // Copy substring
                c.push_back(stringToSplit.substr(startpos, i - startpos));
                startpos = i + 1;
            }
        }
        print(c);
    }
    // Example 2:   Using very old strtok function ----------------------------------------------------------
    {
        // Our string that we want to split
        std::string stringToSplit{ "aaa,bbb,ccc,ddd" };
        Container c{};
        // Split string into parts in a simple for loop
#pragma warning(suppress : 4996)
        for (char* token = std::strtok(const_cast<char*>(stringToSplit.data()), ","); token != nullptr; token = std::strtok(nullptr, ",")) {
            c.push_back(token);
        }
        print(c);
    }
    // Example 3:   Very often used std::getline with additional istringstream ------------------------------------------------
    {
        // Our string that we want to split
        std::string stringToSplit{ "aaa,bbb,ccc,ddd" };
        Container c{};
        // Put string in an std::istringstream
        std::istringstream iss{ stringToSplit };
        // Extract string parts in simple for loop
        for (std::string part{}; std::getline(iss, part, ','); c.push_back(part))
            ;
        print(c);
    }
    // Example 4:   Most flexible iterator solution  ------------------------------------------------
    {
        // Our string that we want to split
        std::string stringToSplit{ "aaa,bbb,ccc,ddd" };
        Container c(std::sregex_token_iterator(stringToSplit.begin(), stringToSplit.end(), delimiter, -1), {});
        //
        // Everything done already with range constructor. No additional code needed.
        //
        print(c);
        // Works also with other containers in the same way
        std::forward_list<std::string> c2(std::sregex_token_iterator(stringToSplit.begin(), stringToSplit.end(), delimiter, -1), {});
        print(c2);
        // And works with algorithms
        std::deque<std::string> c3{};
        std::copy(std::sregex_token_iterator(stringToSplit.begin(), stringToSplit.end(), delimiter, -1), {}, std::back_inserter(c3));
        print(c3);
    }
    return 0;
}
And, with this additional know how, we can come up with a solution that first slits the line, and then compare the specific part of the string to a key.
#include <iostream>
#include <fstream>
#include <string>
#include <vector>
#include <regex>
// Save some typing work and create an alias
using Iter = std::sregex_token_iterator;
const std::string key{ "Back Bay" };
const std::regex separator{ "," };
void readFileVector(std::string& fileName, std::vector <std::string>& fileVector)
{
    // Open input file
    std::ifstream file(fileName);
    // Check, if it could be opened.
    if (!file) {
        //if not, then show error message
        std::cerr << "File could not be opened: " << fileName << std::endl;
    }
    else {
        // File could be opened. Now we want to read line by line
        std::string line{};
        // Read the complete file
        while (getline(file, line)) {
            // Split string into parts
            std::vector part(Iter(line.begin(), line.end(), separator, -1), {});
            // Now the condition
            if ((part.size() > 4) and (part[4] == key))
                fileVector.push_back(line);
        }
    }
}