I'm still learning Japanese. But from what I know I would use shorai. It means more of a near future. mirai is the distant future.

Hi, Jordan's answer is very good. In general, we use しょうらい, however, "my future family" can be translated in both みらいの私の家族 and しょうらいの私の家族 depending on the situation and the person. However, in this case, I would say ホストファミリーのみなさんへ because しょうらいの / みらいの私の家族 is often used for "the family of my fiancee or girl/boyfriend", "my future family in law". If you want to add "future" for your host family, I would say 私のホストファミリーになってくださる家族、私を受け入れてくれるホストファミリー. I hope this was helpful but if you still have doubt, please let me know.

And would I be able to use it like that in a letter? "watashi no shourai no kazoku he"?