Chrome Extension Development - POST to new tab

You can simply combine these two techniques:

1. You may execute JavaScript commands by adding javascript: prefix at your address bar or in href values of <a> tags.
2. Only with JavaScript, you can create a form element and fill it with your data then POST it.

function fakePost() {
    var form = document.createElement("form");
    
    // Create a POST dump at ptsv2.com and change the code
    form.setAttribute("action", "http://ptsv2.com/t/dcgex-1614815819/post");
    form.setAttribute("method", "post");

    var params = { userId: 2, action: "delete" };
    for(var key in params) {
        var hiddenField = document.createElement("input");
        hiddenField.setAttribute("type", "hidden");
        hiddenField.setAttribute("name", key);
        hiddenField.setAttribute("value", params[key]);
        form.appendChild(hiddenField);
    }

    // Appending the form might not be necessary
    document.body.appendChild(form);

    form.submit();
};

const 
    source = fakePost.toString().replace(/(\n|\t)/gm,'').replace(/\s\s/gm,' '),
    url = `javascript:${source}; fakePost();`;

chrome.browserAction.onClicked.addListener(() => chrome.tabs.create({ url }));

Of course, that's just a dirty hack. If you need something more elaborate you can use a XHR Object or @Xan's answer.