GrayHorse

Нужно дождаться выполнения всех fetch().then().then(), собрав их в массиве.

const promises /* !!! */ =  json.dialogs.map((v) => {
    let dialogId = v.dialogId;
    return /* !!! */ fetch(`${this.WebApiHost}/api/qwe/dialogs/${dialogId}/${this.My_Token}`)
    // ....
});
await Promise.allSettled(promises);

А лучше переписать на async/await и не использовать асинхронные функции в map/forEach:

try {
    const resp = await fetch("");
    const json = await resp.json();
    if (!json || json.error === true) {
        console.log("error");
    }

    async function handleDialog(dialog) {
        try {
            const resp = await fetch("");
            const json = await resp.json();
            // ...
        } catch (e) {
            // ...
        }
    }

    const concurrently = false;
    if (concurrently) {
        // [Way 1] Run one by one ---------
        for (const dialog of []) {
            await handleDialog(dialog);
        }
    } else {
        // [Way 2] Run all concurrently ---
        const promises = [];
        for (const dialog of []) {
            promises.push(handleDialog(dialog));
        }
        await Promise.allSettled(promises);
    }
    // The best way: concurrently with limitation
    // @see https://qna.habr.com/q/1105356#answer_2098042

    // ...
} catch (e) {
    // ...
}