Well, normally this isn't a code writing service, but if it helps Perl programmers, I guess it's for a good cause ;-) Will this documentation be public somewhere?

If I understood the C# code correctly, here's one way to do it with HTTP::Tiny and URI (for SSL support, IO::Socket::SSL and Net::SSLeay need to be installed too):

#!/usr/bin/env perl
use warnings;
use strict;
use URI;
use HTTP::Tiny;

my $uri = URI->new('https://api.zerobounce.net/v1/validate');
$uri->query_form(
    apikey    => 'Your Secret Key',
    email     => 'example@example.com',
    #ipaddress => '99.123.12.122',
);
my $response = HTTP::Tiny->new->get( $uri,
    { timeout=>15 } ); # in seconds
die "Failed! $response->{status} $response->{reason}\n"
    unless $response->{success};
my $responseString = $response->{content};
[download]

an online Perl compiler that allows JSON REST Calls

I'd recommend installing Perl locally for testing (and perhaps just pointing the script at a different URL for testing). It comes pre-installed on many *NIX systems, and on Windows, Strawberry Perl makes things fairly easy. All of the aforementioned modules are included in the latest Strawberry Perl release, while on *NIX, some of them may need to be installed.