my $m = WWW::Mechanize->new;

Replace with  WWW::Mechanize->new( qw/ autodie 1 /);

 getstore($url,$filename) or die "Can't download '$url': $@\n";

Get rid of LWP::Simple , replace with  $m->mirror( $url, $filename ); autodie takes care of the dying

 if ( -e -d $name ) {

While you can (since 5.9.1) chain/stack file-test operators in a case like this (it's the quivalent of  -d $name && -e _) , if its a directory, it already exists (otherwise how can it be a directory ), so if you're going to test to see if its a directory, just use -d

Checking if a directory exists before creating the directory is a classic race condition --- some other program could make the directory after your program checks that it doesn't exist, but before it creates it -- this may or may not matter to your program

Since creating a directory is an atomic operation (either succeed or fail), simply try to create the directory -- if it already exists, creation will fail

Here is how I might write that

sub Dirpp {
    use Errno qw/ EACCES /;    # permission denied
    my $newdir = "site00";
    my $made   = 0;
    while ( not $made = mkdir $name, 0755 ) {
        die $! if $! == EACCES;
        $newdir++;
    }
    return $name if $made;     # if we made it, return new name
    return;
}
[download]

This takes advantage of the magical string increment or auto-increment. auto-increment does overflow, so if that isn't desired, then I would write

sub DirPP {
    use Errno qw/ EACCES /; # permission denied
    my $word    = "site";
    my $counter = 1;
    my $name ;
    my $made = 0;
    while(1){
        $name = sprintf '%s%3d', $word, $counter;
        last if not $made = mkdir $name, 0755 ;
        die $! if $! == EACCES;
    }
    return $name if $made;
    return;
}
[download]

Why  while(1) ? Because to me two sprintf looks clumsy

sub DirPo {
    use Errno qw/ EACCES /;    # permission denied
    my $word    = "site";
    my $counter = 1;
    my $made    = 0;
    my $name    = sprintf '%s%3d', $word, $counter;    ## ONE, PEE
    while ( not $made = mkdir $name, 0755 ) {
        die $! if $! == EACCES;
        $name = sprintf '%s%3d', $word, $counter;      ## TWO, EEW
    }
    return $name if $made;
    return;
}
[download]

Also, storing one file per directory is bizzare :)