This idea of risk-taking as single concept is flawed. There are different types of risk-taking in different circumstances with differences in possible outcomes. There are some differences in how men and women take risks, but the differences are not large, there is significant overlap, and of course any one individual will behave according to their specific personality not just as a representative of their sex.
"Is it the case that culturally and socially women aren’t allowed to take risks?"
No, but perhaps culture and society affect the kinds of risks that men and women take, dependent on the risk/reward ratio and the circumstances.
"Or is it that we biologically driven to not?"
No, women do take risks, perhaps a little differently on average than men do. "Biologically" could mean all sorts of things, do you mean due to male v female hormones, or differences in the brains of men/women, or something else?
"If there were no men, would society be as progressive as it is?"
I don't think you mean "progressive" in this sentence. Perhaps you meant "if there were no men, would society have progressed as it has?" If there were no men, then society could be very different, but I think it is clearly unsupported to suggest that society would not have progressed beyond the Stone age.