Is it permissible for my wife to go for Umrah using someone else's money, and is her Umrah valid, and is the money from her sister's husband considered a gift to her?

There is no harm in you allowing your wife to go for 'Umrah at the expense of her sister's husband, and there is no harm in her going, and her 'Umrah is valid, Allah willing. The validity of 'Umrah is not affected by its expenses coming from someone else's money.