I would look at what HTTP requests Curl creates from that command line, and then recreate these HTTP::Requests for use with LWP::UserAgent. If Curl has no convenient way to show what it is sending, likely Wireshark or any other network sniffer will produce the data sent by Curl.
use HTTP::Request::Common;
my $virtual_machines = GET(
'https://api.vps.net/virtual_machines.api10json',
Accept => 'application/json',
Content_Type => 'application/json',
);
# perform request, passing the credentials either in the URL or settin
+g up LWP::UserAgent
...
print $response->as_string;
Setting new values is likely done via POST:
use JSON;
my $new_values => [
machine1 => 100,
machine2 => 200,
machine3 => 300,
machine4 => 400,
];
my $set_virtual_machines = POST(
'https://api.vps.net/virtual_machines.api10json',
Accept => 'application/json',
Content_Type => 'application/json',
Content => [
'magic_json_parameter_name' => encode_json($new_values ),
],
);