chrome.webRequest is helpful but it doesn't let you read the response body in Chrome.
I made an extension that intercepts all web requests using a script that is injected into the page by a content script. My example is here: https://github.com/onhello-automation/onhello/tree/main/app/scripts.
I used https://stackoverflow.com/a/48134114/1226799 to help write this but I corrected some issues in there and simplified it.
Some relevant parts:
manifest.json
    "content_scripts": [
        {
            "matches": [
                "https://example.com/*"
            ],
            "run_at": "document_start",
            "js": [
                "scripts/content_script.js"
            ]
        }
    ],
    "web_accessible_resources": [
        "scripts/injected.js"
    ],
    "permissions": [
        "https://example.com/*"
    ]
scripts/content_script.ts (I use webextension-toolbox to build and I compile TypeScript to JavaScript)
import { browser } from 'webextension-polyfill-ts'
// You can use `browser`/`chrome` here and interact with extension stuff like storage and tabs.
const s = document.createElement('script')
s.src = browser.extension.getURL('scripts/injected.js')
s.onload = async function () {
    (this as any).remove()
};
(document.head || document.documentElement).appendChild(s)
scripts/injected.js:
// You CANNOT use `browser`/`chrome` here and you CANNOT interact with extension stuff like storage and tabs.
const XHR = XMLHttpRequest.prototype
const open = XHR.open
const send = XHR.send
const setRequestHeader = XHR.setRequestHeader
XHR.open = function () {
    this._requestHeaders = {}
    return open.apply(this, arguments)
}
XHR.setRequestHeader = function (header, value) {
    this._requestHeaders[header] = value
    return setRequestHeader.apply(this, arguments)
}
XHR.send = function () {
        
    this.addEventListener('load', function () {
        const url = this.responseURL
        const responseHeaders = this.getAllResponseHeaders()
        try {
            if (this.responseType != 'blob') {
                let responseBody
                if (this.responseType === '' || this.responseType === 'text') {
                    responseBody = JSON.parse(this.responseText)
                } else /* if (this.responseType === 'json') */ {
                    responseBody = this.response
                }
                // Do your stuff HERE.
            }
        } catch (err) {
            console.debug("Error reading or processing response.", err)
        }
    })
    return send.apply(this, arguments)
}