Is there a reason you have to define $wget_url outside of the subroutine? If not, then this change should allow it to work as you desire:

my $testfile="/home/user/symbol_list";

&get_symbol_data($testfile);

sub get_symbol_data {
 open (FILE, $_[0]) || die "Can't open $_[0] $!\n";
 @sym_array = <FILE>;
 close <FILE>;

 foreach $symbol (@sym_array) {
   chomp $symbol;
 
   my $wget_url=qq~/home/user/sym_dir/${symbol}_2013 'http://www.downl
+oadsite.com/${symbol}.dat'~;
   `usr/bin/wget -O $wget_url`;
 } 
}
[download]

The qq~...~ allows all occurrences of $symbol to be properly interpolated, even inside single quotes in a string assignment.