How can I make a script take multiple arguments?

Using the same structure as your original script, you just need to iterate over the $@ array (that's the list of arguments given in the command line):

#!/usr/local/bin/bash
set -e
if [ "$#" -lt 1 ]
then
echo  "Please insert at least one argument"
exit
else
echo -e "\c"
fi


for file in "$@"
do
    if [ -h  "$file" ]
    then
         echo "$file is a symbolic link"
    else    
         echo "$file is not a symbolic link"
    fi
done

A simplified version of the same thing would be:

#!/usr/bin/env bash
[ "$#" -lt 1 ] && printf "Please give at least one argument\n" && exit 
for file 
do
    [ -h "$file" ] && printf "%s is a symbolic link\n" "$file" || 
        printf "%s is not a symbolic link\n" "$file"
done

No one mentioned shift?

if [ x = "x$1" ] ; then
    echo need at least one file
    exit 1
fi

while [ x != "x$1" ] ; do
  if [ -h  "$1" ]; then
    echo "$1 is a symbolic link"
  else    
    echo "$1 is not a symbolic link"
  fi
  shift
done

You can use a for loop to process all files passed to script:

for f do
  if [ -h  "$f" ]; then
    printf "%s is a symbolic link\n" "$f"
  else    
    printf "%s is not a symbolic link\n" "$f"
  fi
done