Is it permissible for the second wife to become pregnant without her husband's permission, knowing that he refuses to have more children because he has many children from his first wife, and she desires more and fears missing the opportunity to conceive?

Childbearing is a right for both spouses. The husband does not have the right to prevent his wife from it without a legitimate Shariah justification. The fact that he has other children does not justify him preventing his wife from becoming pregnant, and sustenance is in the hands of Allah. It is advisable to try and convince the husband to allow pregnancy. If he insists [on his refusal], then there is no blame upon the wife for conceiving without his knowledge if possible.