You say

I've tried reading it in and setting up for loops and such, but I cannot get anything that works.

What have you tried? What didn't work? What errors did you get? How do you know it didn't work? Post some code (wrapped in <code> tags), so we can help guide you to a working tool. This is not a code writing service.

Update: Now that you have updated your post with code, I can comment. First, the posted code with the posted input file yields the warnings:

Use of uninitialized value $_ in pattern match (m//) at fluff.pl line 
+26, <IN> line 9.
Argument "1 1 1" isn't numeric in numeric ne (!=) at fluff.pl line 26,
+ <IN> line 9.
Use of uninitialized value $_ in pattern match (m//) at fluff.pl line 
+26, <IN> line 9.
Argument "2 2 2" isn't numeric in numeric ne (!=) at fluff.pl line 26,
+ <IN> line 9.
Use of uninitialized value $_ in pattern match (m//) at fluff.pl line 
+26, <IN> line 9.
Argument "3 3 3" isn't numeric in numeric ne (!=) at fluff.pl line 26,
+ <IN> line 9.
Use of uninitialized value $_ in pattern match (m//) at fluff.pl line 
+26, <IN> line 9.
Argument "" isn't numeric in numeric ne (!=) at fluff.pl line 26, <IN>
+ line 9.
Use of uninitialized value $_ in pattern match (m//) at fluff.pl line 
+26, <IN> line 9.
Argument "4 4 4" isn't numeric in numeric ne (!=) at fluff.pl line 26,
+ <IN> line 9.
Use of uninitialized value $_ in pattern match (m//) at fluff.pl line 
+26, <IN> line 9.
Argument "5 5 5" isn't numeric in numeric ne (!=) at fluff.pl line 26,
+ <IN> line 9.
Use of uninitialized value $_ in pattern match (m//) at fluff.pl line 
+26, <IN> line 9.
Argument "6 6 6" isn't numeric in numeric ne (!=) at fluff.pl line 26,
+ <IN> line 9.
[download]

This is because you've used numeric unequal (!=, Equality Operators) in place of the negative binding operator (!~, Binding Operators). This is problematic because without binding, the regular expression is tested against your uninitialized magic variable $_. What you actually meant is not that the line doesn't contain any whitespace, but rather that the line contains a character that is not whitespace. You can achieve this using the \S character class, so the block becomes:

  elsif ($line =~ /\S/) {
      push (@arr, "$line\n");
      next;
  }
[download]

If we run this, we get your intended outout, but as you say, are missing one output file. This can be resolved by adding a final call to your create_file sub, so the final, functional version would be:

#!/usr/bin/perl
use strict;
use warnings;

my $infile = 'roegen6.vect';
my $count = 1;
my $outfile = "$infile-section_$count.vect";
my @arr;

sub create_file {
    open(OUT,">$outfile") or die "Error with outfile: $!\n"; 
    print OUT @arr;
    close(OUT);
    @arr=();
    $count++;
    $outfile="$infile-section_$count.vect";
}

open(IN,$infile) or die "Error with infile $infile: $!\n";
my @data=<IN>;
close(IN);

 foreach my $line (@data) {
  chomp($line);

  if ($line =~ /VECT/) {
      push (@arr, "$line\n");
      next;
  }
  elsif ($line =~ /\S/) {
      push (@arr, "$line\n");
      next;
  }
  else {
    push (@arr, "$line\n");
    create_file();
  }
 }
create_file if @arr;
[download]

Note I've put a conditional on the final output, so it will only write if your buffer has content. Not quite how I would have written it from scratch, but it works.