Letter to host family

The host family that I am staying with has just written me an email and I have written a response to it in Japanese. (It is my first time talking to them) But I am not sure if it is okay or not or if it has mistakes. If someone could tell me if it is okay or how to improve it that would be much appreciated.

ホストファミリーのみなさんへ
はじめまして、(name)です。
皆さんに会えることを、私も楽しみにしています。
サポートをいただけるとのこと、本当にありがとうございます。
日本語はとくいではありませんでもじょうたつするようにがんばります。
ホームステイをさせてくれてありがとうございます。
よろうしくおねがいします。

Also, she has written a separate part at the end of the email to my parents saying that she will take good care of me. How do I write that they want to thank her and should I put it separately at the end of the email?

It looks very good, just a typo in the last sentence:
よろうしくおねがいします。
It should be
よろしくおねがいします。

About adding that your parents thank the host family, maybe you could add one sentence before the last sentence.

Thank you for your help!
Also, what do you think the best way to thank them on behalf of my parents in Japanese is?

It looks mostly okay.

I would omit the second last sentence
「ホームステイをさせてくれてありがとうございます。」
because it sounds redundant and is needles to say when you reply to the host.

Also, I would change some words.

皆さんと会う日を、私も楽しみにしています。
サポートしていただけるとのこと、
本当にありがとうございます。
日本語はとくいではありませんが、がんばります。
よろしくおねがいします。

「皆さんと会う」 sounds better because you say 「私も」[ I too ].
「でも」 is a word mainly for speaking, in a casual way.

what do you think the best way to thank them on behalf of my parents in Japanese is?

How about citing your parents' words to encourage you?

One thing your host family would worry is that your parents may miss you very much.
If your parents have seen you off agreeably,
your host family would be relived to hear it, I suppose.