My concern is this. With 7 or more characters, the possible letter combinations are in the thousands.. and then I have to check each combination to verify that it's a word. I'd like to do this using CGI but I fear I won't be able to get it fast enough where it won't time out.

You're thinking about this in the wrong direction. If you want to preprocess your dictionary like this (start with a word and then check whether any of its permutations are words), it will take a very long time. In fact, you will be checking the dictionary for every possible combination of letters. Since there are far fewer words in the dictionary than possible combinations of letters, you should look in the other direction: for each word in the dictionary, normalize it (to its sorted form), and then store it in a hash keyed on the normalized version.

my @words = do {
    open my $fh, "<", "/usr/dict/words" or die;
    <$fh>;
};
chomp @words;

sub normalize {
    join "", sort split //, $_[0];
}

my %dict;
push @{ $dict{normalize($_)} }, $_
    for @words;
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This will give you a structure %dict that looks like this:

  abd => ["bad", "dab"],
  aepr => ["pear", "reap", "rape"]
  ...
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Then if you want to find the words that are permutations of $foo, you look at the arrayref $dict{normalize($foo)}. You should also save this processed dictionary file somewhere, so you don't have to do all this work twice.

Is map the best way of going about this? I am not too familiar with map and this ability. And I really don't understand the logic of how to use the anagrams or why it would be faster.

Map is just a looping construct like foreach, while, etc.. It can't do anything magical that cannot be done with another type of programming construct. You can write both inefficient and efficient code with map. It will not write your algorithms for you. If you don't understand it, don't use it.