I am adding one more method which uses HTML::TreeBuilder::XPath to select what you want using an XPath expression. Mainly for sentimental reasons because TreeBuilder helped me out in many a cases, but also because XPath expressions are useful to know and use.

A word of warning: HTML::TreeBuilder seems to ignore unknown-to-it html tags by default, e.g. HTML5 tags like section. A quick but dirty fix is to instruct it to lax a bit, see Issues using HTML::TreeBuilder::XPath and the HTML5 <section> tag

Here is example code tested with your html string:

#!/usr/bin/env perl

use strict;
use warnings;

use HTML::TreeBuilder::XPath;

my $html = '<select class="c-form__select" id="radius" name="radius" a
+ria-label="Choose a distance from your postcode"><option value="">Dis
+tance (national)</option><option value="1">Within 1 mile</option><opt
+ion value="5">Within 5 miles</option><option value="10">Within 10 mil
+es</option><option value="15">Within 15 miles</option><option value="
+20">Within 20 miles</option><option value="25">Within 25 miles</optio
+n><option value="30">Within 30 miles</option><option value="35">Withi
+n 35 miles</option><option value="40">Within 40 miles</option><option
+ value="45">Within 45 miles</option><option value="50">Within 50 mile
+s</option><option value="55">Within 55 miles</option><option value="6
+0">Within 60 miles</option><option value="70">Within 70 miles</option
+><option value="80">Within 80 miles</option><option value="90">Within
+ 90 miles</option><option value="100">Within 100 miles</option><optio
+n value="200">Within 200 miles</option></select>';

my $TB = HTML::TreeBuilder::XPath->new(
    # if you have html5 with tags unknown to 'HTML::TreeBuilder' then 
+uncomment following
    # ignore_unknown => 0,
);
$TB->parse($html) || die 'TB->parse()';

#my $xpath = '//form[@name="searchVehicles"]//select[@name="radius"]/o
+ption[@value!=""]/@value';
my $xpath = '//select[@name="radius"]//option[@value!=""]/@value';

my @values = $TB->findvalues($xpath);
print "found ".scalar(@values)." option values:\n";
foreach my $avalue (@values){
    print "value: '".$avalue."'\n";
}
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p.s. Personally, I would use a package which can cope better with latest HTML standards. XPath for me is appealing, say like a regex is. However, if what you are building is going to be a long term, expanding project you need to maintain - easily - your XPaths whenever site's web design changes. Maybe then Mojo::DOM (which I have not tried) is the way to go.