"If a woman does not feel that she could allow her husband to marry
more than one woman at the same time, then Islam allows her the right
to refuse him permission to do this at the outset of their marriage,
however, she must indicate this preference in the Prenuptial Agreement
or she will forfeit this right under the Islamic Law. If she is
uncertain as to whether or not she will be opposed to her husband
marrying a second wife later on, then she could include that in the
agreement and thus make it binding upon her husband that he must
consult her at that time and that he must then abide by her wishes. To
say nothing, however, could possibly invite more pain than gain as far
as her desires are concerned. In the West, polygamy is illegal. Even
so, the woman may still request that her husband not marry a second
wife, and put this in the contract. This sort of request would be
considered spurious in the Prenuptial Contract because men in the west
are already forbidden polygyny. Nonetheless it might still prove to be
a useful addition to the contract at this time because later on the
couple might possibly move to a country where polygyny is legal."
You can look more in here
There is and other view, it says:
"because of the harm and mess that inevitably results:
a) the harm to the first wife;
b) the troubles with the second wife when the first is upset;
c) the harm of not giving both their legal, emotional, and material
d) the harm to family relations;
e) and, also vitally, the harm to one’s children…"
You can find resource in here
Allah Most High says:
“If you fear that you shall not be able to deal justly with the orphans, marry women of your choice, two, three, or four, but if you fear that you shall not be able to deal justly (with them), then only one…(Surah al-Nisa, 3)."
So at this conditions,in my opinion; if first wife refuse to allow, husband can't have permission.