Looking at the code of Mojo::DOM, it doesn't look like it's directly supported. But luckily it's not too difficult to add (you can of course put the package into its own .pm file):

Update: I've modified the methods so that they return a nested set of Mojo::Collection objects of the callback results, so that walk is kind of like a tree-based map.

Update 2: For an even more refined version, see here.

use Mojo::Base -strict;
use 5.014;
use Mojo::UserAgent;
use Mojo::DOM;
use Mojo::Util qw/dumper/;

package Mojo::DOM::Role::TreeWalker {
    use Mojo::Base -strict;
    use Role::Tiny;
    use Mojo::Collection;
    sub walk { $_[0]->_walk($_[1], 0) || Mojo::Collection->new }
    sub _walk {
        my ($self, $cb, $depth) = @_;
        my $c = Mojo::Collection->new;
        { local $_ = $self; push @$c, $cb->($self, $depth++); }
        my $rv = $self->child_nodes->map('_walk', $cb, $depth);
        push @$c, $rv if @$rv;
        @$c ? $c : ();
    }
    sub walk_text {
        my ($self, $cb) = @_;
        $self->walk(sub { $_->type eq 'cdata' || $_->type eq 'raw'
            || $_->type eq 'text' ? $cb->(@_) : () });
    }
}

my $ua = Mojo::UserAgent->new( max_redirects => 3 );
my $res = $ua->get('http://www.spacex.com/webcast')->result;
die $res->message unless $res->is_success;
my $dom = $res->dom;

$dom->find('script, style')->map('remove');

my $texts = $dom->with_roles('+TreeWalker')->walk_text(sub {
    $_->content=~/\S/ ? $_->content=~s/^\s+|\s+$//gr
        : () })->flatten;
print dumper $texts;

__END__

bless( [
  "STARLINK MISSION | SpaceX",
  "Jump to navigation",
  "Falcon 9",
  "Falcon Heavy",
  "Dragon",
  "Starship",
  "Updates",
  "About SpaceX",
  "Careers",
  "Shop",
  "You are here",
  "Home",
  "STARLINK MISSION",
  "On Wednesday, April 22 at 3:30 p.m. EDT, or 19:30 p.m. UTC, SpaceX 
+launched its seventh Starlink mission. Falcon 9 lifted off from Launc
+h Complex 39A (LC-39A) at NASA\x{2019}s Kennedy Space Center in Flori
+da.",
  "Falcon 9\x{2019}s first stage previously supported Crew Dragon\x{20
+19}s first flight to the International Space Station, launch of the R
+ADARSAT Constellation Mission, and the fourth Starlink mission. Follo
+wing stage separation, SpaceX landed Falcon 9\x{2019}s first stage on
+ the \x{201c}Of Course I Still Love You\x{201d} droneship, which was 
+stationed in the Atlantic Ocean. Falcon 9\x{2019}s fairing previously
+ supported the AMOS-17 mission.",
  "You can watch a replay of the launch below and learn more about the
+ mission",
  "here.",
  "|",
  "Twitter",
  "YouTube",
  "Flickr",
  "Instagram",
  "Privacy",
  "\x{a9} 2020 Space Exploration Technologies Corp."
], 'Mojo::Collection' )
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