You can just ls it, then check the exit code of ls. If it's 0 then the file exist, otherwise it doesn't.

$ ls wfh.sh
wfh.sh
$ echo $?
0
$ ls a
ls: cannot access 'a': No such file or directory
$ echo $?
2

In your case, you can write:

ls ${INPUT_DIRECTORY}*.tgz > /dev/null 2>&1

if [ $? -eq 0 ] ; then
  echo "exist!"
else
  echo "not!"
fi

Also, you should change your globbing to ${INPUT_DIRECTORY}*.tgz instead of ${INPUT_DIRECTORY}*tgz since it may unintentionally match something like filletgz

For more readable script, you can also write:

if ls ${INPUT_DIRECTORY}*.tgz > /dev/null 2>&1 ; then
  echo "exist!"
else
  echo "not!"
fi

Here’s a suggestion for a test, in pure shell:

for i in "$input_dir"/*.tgz "$input_dir"/.*.tgz; do
    test -e "$i" && break
done

This returns success or failure, depending on if .tgz files exist. You can obviously remove the second pattern if you want to ignore hidden files.

As a function/example:

suffix_exists ()
{
    for i in "${2:-.}"/*"$1" "${2:-.}"/.*"$1"; do
        test -e "$i" && return 0
    done
}

if suffix_exists .tgz "$input_dir"; then
    echo "$input_dir contains .tgz archives"
else
    echo "$input_dir does not contain .tgz archives"
fi

It uses the current directory if none is provided.

As mentioned in another answer, you might be approaching your task in the wrong way (perhaps post your objective). Instead of detecting .tgz archives in the directory, you could run a command on any archives that do exist, using find. For example extracting them:

find "$input_dir" -mindepth 1 -maxdepth 1 \
     -type f -name '*.tgz' -exec tar xf {} \;

Or moving them somewhere:

find "$input_dir" -mindepth 1 -maxdepth 1 \
     -type f -name '*.tgz' -exec mv -i {} /my/archives \;