You're trying to match /\bmy term 3\b/ against each "word" (groups of non-space characters) of line.
Even if the line contains my term 3,
'my'   =~ /\bmy term 3\b/ will be false,
'term' =~ /\bmy term 3\b/ will be false, and
'3'    =~ /\bmy term 3\b/ will be false.

Just eliminate the foreach my $word (@word_array) loop, leaving its body in place.

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Other problems:

$word_count isn't reset to 0 for each term like it should.

You don't check if the two arguments were provided.

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Some style tips for better readability:

@array_1 and @array_2 are meaningless names.

$pathway_abstracts and $terms are not much better. There's not even a hint that these are mearly the names of the file that contain the information instead of the information itself.

for (my $j = 0; $j < @array_2; $j++)
is less readable than
for my $j (0 .. $#array_2)
In this case, you don't even need $j, so I'd recommend
for my $term (@array_2)

Same for the $i loop.
for my $line (@array_1)

The second chomp does nothing. The first chomp already did the deed. Remember, $key in for my $key (@array) is a alias to the array element. Any change to $key will affect the array element to which it is linked.

The placement of chomp is odd. I'd move it to where the array is read in. Change
my @array_2 = <IN2>;
to
chomp( my @array_2 = <IN2> );

Why have two loops going iterating over @array_2? Move
$term_score{$key} = 0;
into the second loop.

$x = $x + $y;
can be written much more simply as
$x += $y;

scalar isn't needed when already in scalar context.

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Simplified solution:

#!/usr/bin/perl -w

use strict;
use warnings;

use Getopt::Long qw( GetOptions );

# Get user input.
my $abstracts_file;
my $terms_file;
GetOptions(
   "pathway_abstracts=s" => \$abstracts_file,
   "terms_file=s"        => \$terms_file,
);
# ...Needs error checking here...

# Load pathway abstract.
my $file;
{
  open(my $fh, '<', $abstracts_file)
    or die("Unable to open abstracts file \"$abstracts_file\": $!\n");
  local $/;
  $file = <$fh>;
}

# Load terms into array.
my @terms;
{
  open(my $fh, '<', $terms_file)
    or die("Unable to open terms file \"$terms_file\": $!\n");
  chomp( @terms = <$fh> );
}

print("Term\t| ");
print("Number\t| ");
print("Frequency\n");
print("----------------------------------------------------\n");

# Find out how many words are in abstracts.
my $word_count = () = split(' ', $file);

for my $term (@terms) {
  # Count the number of times the search term matches.
  my $score = () = $file =~ /\b\Q$term\E\b/g;
  my $freq = $score / $word_count;
  print("$term\t$score\t$freq\n");
}
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Update: Added more tips.
Update: Added solution.