Well, given the the array ref return by $entry->get(...), there's not much that can be done to simplify it further.

If you wanted to go for gold on Aristotles patented scale of Yukiness, you could reduce it somewhat more

for my $e ( $searchresult->entries ) {
    my ($r, $uid) = ($e->get('uid'),  $r ? $r->[0] : ());
    
    print "*** No uid for the following  entry:\n" , Dumper \$e
    and next unless $uid;
        
    ($r, $uids{$uid}{$_}) = ($e->get($_), $r ? $r->[0] : '' ) for @var
+s;
}
[download]

I'll probably get drummed out of the PerlMonks Code-Scribes Guild for even joking about that......but then again, I was never invited to become a member, so what the hay:)

However, if its in your power to arrange for $entry->get(..) to return a list instead of an array reference, then the code gets much cleaner....

for my $entry ( $searchresult->entries ) {
    my ( $uid ) = $entry->get( 'uid' );
    
    print "*** No uid for the following  entry:\n" , Dumper \$entry
        and next unless $uid;
        
    ( $uids{$uid}{$_} ) = $entry->get($_) for @vars;
}
[download]

which is much nicer I think, but can you change get()?

If you can, then it would be even better to have it use wantarray to test for the return context and only return the first value of the array (or undef or '') if called in a scalar context (as this seems to be all you want) and the whole array (or ()) in a list context. Then you can remove the parens around the assignments in the last example and it get a little cleaner still.

Note: Obviously none of the examples have been tested in context. I've checked individual bits, but...