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;
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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 ),
    ],
);
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