Perl - Multiple condition if statement without duplicating code?

I don't recommend storing passwords in a script, but this is a way to what you indicate:

use 5.010;
my %user_table = ( tom => '123!', frank => '321!' );

say ( $user_table{ $name } eq $password ? 'You have gained access.'
    :                                     'Access denied!'
    );

Any time you want to enforce an association like this, it's a good idea to think of a table, and the most common form of table in Perl is the hash.

if (   ($name eq "tom" and $password eq "123!")
    or ($name eq "frank" and $password eq "321!")) {

    print "You have gained access.";
}
else {
    print "Access denied!";
}

Simple:

if ( $name eq 'tom' && $password eq '123!'
    || $name eq 'frank' && $password eq '321!'
) {

(use the high-precedence && and || in expressions, reserving and and or for flow control, to avoid common precedence errors)

Better:

my %password = (
    'tom' => '123!',
    'frank' => '321!',
);

if ( exists $password{$name} && $password eq $password{$name} ) {