First a few suggestions:

• Always use strictures (use strict; use warnings;)
• use the three parameter version of open and always check the result.
• use lexical file handles (declared with my)
• use the Perl version of the for loop, not the C version
• avoid opening and closing the same file multiple times

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

# Fake up a couple of files
my $file_a = <<TXT;
a1\tb1\tc1\td1
a2\tb2\tc2\td2
a3\tb3\tc3\td3
TXT
my $file_b = <<TXT;
starting form the top of the file I need 1. to get the the value in th
+e 3rd
column (c1)

2. search in a second file (file_B.txt, not tab delimited and quite me
+ssy) all
the matches for it.

3. when a match is found, I would like to append to the current value 
+(c1), the
value of 4th column (d1) in file_A.txt, separated by a space.

4. go back to the first file (file_A.txt), get the the value in the 3r
+d column
in the second row (c2) and do another round of search and insert the v
+alue of d2
in the second file (file_B.txt).
TXT

# Now the 'real' work - \$file_b treats $file_b as a file
open my $inB, '<', \$file_b or die "Can't open file_b: $!";
my $fileBStr = do {local $/; <$inB>}; # Slurp in all of file_b
close $inB;

open my $inA, '<', \$file_a or die "Can't open file_a: $!";

while (<$inA>) {
    chomp;
    
    my @parts = split /\t/;
    
    next if @parts < 4;
    
    $fileBStr =~ s/\b $parts[2] \b/$parts[2] $parts[3]/xgm;
}

close $inA;

print $fileBStr;
[download]

Prints:

starting form the top of the file I need 1. to get the the value in th
+e 3rd
column (c1 d1)

2. search in a second file (file_B.txt, not tab delimited and quite me
+ssy) all
the matches for it.

3. when a match is found, I would like to append to the current value 
+(c1 d1), the
value of 4th column (d1) in file_A.txt, separated by a space.

4. go back to the first file (file_A.txt), get the the value in the 3r
+d column
in the second row (c2 d2) and do another round of search and insert th
+e value of d2
in the second file (file_B.txt).
[download]

Reading the file you are editing into memory is fine unless its size is hundreds of megabytes. For very large files you probably need to turn the loop inside out - read all the edit information from file a and store that in memory, then read file b a line at a time and apply all the edits to the current line before saving it and moving on to the next.