I think what you're looking for is this:

use strict;
use warnings;
use Tk; 


# User-defined
my @menubutton_items = (
    'supplier 1', 'supplier 2', 'supplier 3', 'supplier 3',
);


# Main program
my $chosen_item = "";
my $mw = new MainWindow;
$mw->geometry("400x400+200+0");

my $frm_name = $mw-> Frame(
      -relief=>'groove',
      -borderwidth=>5,
      -background=>'black'
)->pack(-side =>'top', -fill=>'x');

my $expensemb = $frm_name->Menubutton(
    -tearoff => 0,
    -text    =>"Eksodo",
)->pack(-side=>'left',-anchor=>'nw');

# Pack each item into the Menubutton
foreach my $item (@menubutton_items) {
    $expensemb->command(-label => $item, -command => sub { addtext($it
+em) });
}

my $costlabel = $frm_name -> Label(-text=>"Timh",
                                   -background => 'red')
                          -> pack(-side=>'left', -anchor=>'nw', -padx=
+>70);
              
my $frm_name2 = $mw -> Frame(
    -relief=> 'groove',
    -borderwidth  => 5,-background   => 'black')
    ->pack(-side => 'top',-fill => 'x');

my $expenseentry = $frm_name2->Entry(-width=> 10, -textvar => \$chosen
+_item)
                             -> pack(-side =>'left',-anchor =>'nw');

MainLoop;


# Subroutines
sub addtext {
    my ($item) = @_;
    $chosen_item = $item;
}
[download]

Using the command -textvar => \$chosen_item links whatever is assigned to $chosen_item to the Entry widget.  (You might even consider maken it a Label widget, so that the user can't change its contents; you could use the -relief and -border options to make it look just like an Entry widget, but without being directly modifiable).

The line:

    $expensemb->command(-label => $item, -command => sub { addtext($it
+em) });
[download]

packs each text item into the Menubutton, and associates that text with a callback to addtext.  You could even certainly do:

    $expensemb->command(-label => $item, -command => sub { $chosen_ite
+m = $item });
[download]

if you wanted to do nothing other than assign the Entry text, but I made it into a separate subroutine in case you need to take other actions when the user makes a selection.