It isn't supposed to respond. It is doing exactly what it should: it reads the input and assigns it to the variable fileType. However, you then have a while loop which is checking the value of a variable that never changes:

while [[ "$fileType" != "EXIT" ]];

Since the value of $fileType is set only once and before the while loop, that while becomes an infinite loop unless you pass EXIT on the first try. In any case, your script wouldn't work anyway since the output of grep -q pattern file will never be 1. In fact, it will always be empty since that's what grep -q does. I suspect you meant to check the exit status of the command but, even that is pointless: there is no benefit in reading the entire file once with grep -q only to then read it all over again with grep -F.

Here's a working version of your function:

function findFiles() {

  while [[ "$fileType" != "EXIT" ]];

  do

    echo "Enter file type to search for (include '.') || Type EXIT to terminate program "

    read fileType

    grep -F "$fileType" index.html >> foundFiles.txt

  done

     rm index.html

 };

Or, if you want to avoid creating an empty foundFiles.txt if no matches were found, you can try:

function findFiles() {

  while [[ "$fileType" != "EXIT" ]];

  do

    echo "Enter file type to search for (include '.') || Type EXIT to terminate program "

    read fileType

    if grep -qm1 "$fileType" index.html; then

      grep -F "$fileType" index.html >> foundFiles.txt

    fi

  done

     rm index.html

 };

The -m ensures the grep exits after the first match so you don't need to read the whole file twice. Note how I am not using the command substitution ($(command)) syntax since I am checking that the command was successful, and not trying to use its output.