How to get the current tab's history in a Web Extension in Firefox?

No. You cannot ask for the list for a certain tab by default.

You can, however, listen for the tab events onUpdated, onCreated etc. Using the tabId which stays the same, you can keep a list of URLs in a background script (background.js) which is always running if the addon is enabled.

You would do it like this:

let arr=[]; // At the top of background.js
browser.tabs.onCreated.addListener(handleCreated); // Somewhere in background.js

function handleCreated(tab) {
     let tabId = tab.id;
     if(arr[tabId]==null) arr[tabId] = [];
     arr[tabId].push(url);
}

function getHistoryForCurrentTab(){
    function currentTabs(tabs) {
        // browser.tabs.query returns an array, lets assume the first one (it's safe to assume)
        let tab = tabs[0];
        // tab.url requires the `tabs` permission (manifest.json)
        // We will now log the tab history to the console.
        for(let url of arr[tab.id]){
            console.log(url);
        }
   }

   function onError(error) {
       console.log(`This should not happen: ${error}`);
   }

   browser.tabs.query({currentWindow: true, active: true}).then(currentTabs, onError);
}

The above code is a proof of concept. Some improvements you will need to consider: implement onClosed which resets the tab history for that id (arr[tabId] = null), implement onUpdated (will be needed for sure, same logic as in handleCreated).

Links:

• https://developer.mozilla.org/en-US/Add-ons/WebExtensions/API/tabs