Whether you should suspect simultaneity between $y \equiv$ "IPO first-day market returns" and $x \equiv$ "IPO underwriters" depends on your intuition and economic theory. If there are papers which consider a simultaneity problem the authors will usually provide good reasons for this. Otherwise you would not want to sacrifice the efficiency of the OLS estimator for inefficient but consistent IV models.
As concerns your question about the insignificance of the (potentially) endogenous variable this can also be due to small sample size, too little variation in $x$ or too few other explanatory variables. It is possible that the simultaneity bias contributes to this insignificance. To see this write the model as
$$y = \beta_{0} +\beta_{1}x + u$$
In the case of simultaneity, you could also regress
$$x = \delta_{0} + \delta_{1}y + v$$
This means that a high $u$ implies a high value for $y$ which again increases the value for $x$, leading to a correlation between $u$ and $x$. This violates the important OLS assumption that $E(xu)=0$ and you will get inconsistent estimates. Instead you have:
$$E(xu) = E((\delta_{0} + \delta_{1}y + v)u) = \delta_{1}E(yu) +E(uv)$$
Then your estimate of $\beta_{1}$ will be the "true" value plus the simultaneity bias:
$$\widehat{\beta_{1}} = \beta_{1} + \frac{Corr(xu)}{Var(x)}$$
If the bias is positive and large (all else equal!), you get a large value for your estimate of $\widehat{\beta_{1}}$. Dividing the coefficient by the corresponding t-statistic you get the standard error from which you calculate statistical significance. In this sense it is possible that your OLS estimate is insignificant due to the bias problem. The difficulty is that IV/2SLS are far less efficient than OLS and the standard errors are usually larger. Unless the bias reduction is large enough relative to the increase in the standard errors, IV/2SLS are unlikely to be helpful.
