There is no storage associated with file:// URLs. If you can set up a simple local server, or host it on one of the various services with a free tier, that would probably be best.

To do it with just a file, you'll have to have the user explicitly load and save the file:

• Have your page load a local file by having the user drag-and-drop or select it via an input type="file" element.
• Allow the user to save it by generating the file text in memory and then offering it for download so they can save it locally.

This isn't great user experience, of course.

Something along these lines (CodeSandbox link — sadly, Stack Snippets don't allow download):

<div>
    <label>
        Load list:
        <input id="file-load" type="file" />
    </label>
</div>
<input id="btn-save" type="button" value="Save List" />
<ul id="list"></ul>
<div>
    <label>
        Add:
        <input id="item-text" type="text" width="20" />
    </label>
    <input id="btn-add" type="button" value="Add" />
</div>

const listElement = document.getElementById("list");
const fileInput = document.getElementById("file-load");
const saveButton = document.getElementById("btn-save");
const addButton = document.getElementById("btn-add");
const itemText = document.getElementById("item-text");

function addListItem(text) {
    const li = document.createElement("li");
    li.textContent = text;
    listElement.appendChild(li);
}

fileInput.addEventListener("change", (e) => {
    e.currentTarget.files[0].text()
    .then(json => {
        const list = JSON.parse(json);
        for (const element of list) {
            addListItem(element);
        }
    })
    .catch(error => {
        alert("Error loading list file!");
    });
});

saveButton.addEventListener("click", () => {
    const list = Array.from(
        listElement.children,
        li => li.textContent
    );
    const json = JSON.stringify(list);
    const blob = new Blob([json], {
        type: "application/json"
    });
    const objUrl = URL.createObjectURL(blob);
    const link = document.createElement("a");
    link.download = "list.json";
    link.href = objUrl;
    document.body.appendChild(link);
    link.click();
    setTimeout(() => {  // Older Firefox needed this
        document.body.removeChild(link);
    }, 100);
});

addButton.addEventListener("click", () => {
    const text = itemText.value.trim();
    if (text) {
        addListItem(text);
    }
    itemText.value = "";
    itemText.focus();
});

That's just maintaining a list of strings, but you can make it a list of objects instead, so you can have your checked flag, etc.)