means _to see to it_. It is often used with adverbs, without mentioning explicitly what is done (in this sense, it is close to _to manage_ ).
From the linked entry:
>
>
> I'll see to it.I'll take care of it somehow.
It can be used also like _to see to it that ..._ , which is the case for the sentence in question.
Literally
> ****
can be translated to
> I **will have someone see to it that (I) can (/have the permission to) be back** in Okino's house till the end of the vacation of the brothers.
Practically your understanding is correct, but I made the literal translation for .
* * *
In grammar terms, understanding as _so that_ and as _make arrangements_ should clarify.