Hmm, well it is still using a standard form control:
<form class="web-save-form" id="web-save-form" name="wwmform_save" action="/save" method="POST" _lpchecked="1">
<input type="text" class="form-control web-save-url-input web_input web_text" id="web-save-url-input" name="url" placeholder="https://" autofocus="">
<div>
<input type="checkbox" name="capture_outlinks" id="capture_outlinks">
<label for="capture_outlinks">Save outlinks</label>
</div>
<div>
<input type="checkbox" name="capture_all" id="capture_all" checked="">
<label for="capture_all">Save error pages (HTTP Status=4xx, 5xx)</label>
</div>
<div>
<input type="checkbox" name="capture_screenshot" id="capture_screenshot">
<label for="capture_screenshot">Save screen shot</label>
</div>
<div id="wm-save-option" style="display: none">
<input type="checkbox" name="wm-save-mywebarchive" id="wm-save-mywebarchive">
<label for="wm-save-mywebarchive">Save also in my web archive</label>
</div>
<input type="submit" class="web-save-button web_button web_text" value="SAVE PAGE" style="margin: 16px 0;">
</form>
So it appears to be using a POST to http://web.archive.org/save and the other fields will be encoded into the body of the POST response.
There are tools that will let you do test GET/POST submissions and you could try that to see if it works though it isn't clear without trawling through a lot of code whether the page is doing anything clever with JavaScript but it doesn't look like it to me.
So in theory, you should be able to use Node-RED with an http-request node set to POST and passing the parameters on the payload. Here is an example just using Node-RED:
[{"id":"c93cb35b.9df52","type":"http request","z":"5cf69d1b.d88c14","name":"","method":"POST","ret":"txt","paytoqs":true,"url":"http://127.0.0.1:1880/nr/testpost","tls":"","persist":false,"proxy":"","authType":"","x":710,"y":960,"wires":[["3f52e332.e2e23c"]]},{"id":"f193ee3d.cb73c","type":"http in","z":"5cf69d1b.d88c14","name":"","url":"/testpost","method":"post","upload":false,"swaggerDoc":"","x":340,"y":1040,"wires":[["f2e9ec2c.c6bfd","34707403.93638c"]]},{"id":"f2e9ec2c.c6bfd","type":"http response","z":"5cf69d1b.d88c14","name":"","statusCode":"","headers":{},"x":800,"y":1040,"wires":[]},{"id":"34707403.93638c","type":"debug","z":"5cf69d1b.d88c14","name":"","active":true,"tosidebar":true,"console":false,"tostatus":false,"complete":"true","targetType":"full","x":770,"y":1100,"wires":[]},{"id":"bcc33041.382ea","type":"inject","z":"5cf69d1b.d88c14","name":"","topic":"","payload":"","payloadType":"date","repeat":"","crontab":"","once":false,"onceDelay":0.1,"x":340,"y":960,"wires":[["3a9686e5.13200a"]]},{"id":"3a9686e5.13200a","type":"change","z":"5cf69d1b.d88c14","name":"","rules":[{"t":"set","p":"payload","pt":"msg","to":"{\"url\":\"http://example.com\",\"capture_outlinks\":false,\"capture_all\":true,\"capture_screenshot\":false,\"wm-save-mywebarchive\":false}","tot":"json"}],"action":"","property":"","from":"","to":"","reg":false,"x":520,"y":960,"wires":[["c93cb35b.9df52"]]},{"id":"3f52e332.e2e23c","type":"debug","z":"5cf69d1b.d88c14","name":"","active":true,"tosidebar":true,"console":false,"tostatus":false,"complete":"true","targetType":"full","x":870,"y":960,"wires":[]}]