Is it possible for a regression to have the right functional form but still suffer from omitted variable bias? Suppose that
$$ Y = b + aX + e$$
where you know that $E[Y|X] = b + aX$.  Is it true that the model cannot suffer from omitted variable bias?  If this is true, then it follows that omitted variable bias can always be considered as a funcional form bias (namely, changing the functional form $f(X)$ will remove the bias.)
Here is a proof of why I think this is true.  Since $b + aX = E[Y|X]$, we have that
  $$ E[e|X] = E[Y-b-aX|X]= E[Y - E[Y|X] |X] = E[Y|X]  - E[E[Y|X] |X] = E[Y|X] - E[Y|X] = 0.$$
Therefore, $e$ and $X$ are not correlated and there is not omitted variable bias.
 A: Adding or deleting variables in the RHS of the equation is automatically a "change of the functional form", of any function, not just in the case of a stochastic specification. After all a function is determined by its domain and its functional form. So I don't see what we gain by "realizing" that this is so. By definition, if the functional form is "right", there are no omitted variables.  
Moreover, the situation can be more complex. The specification may suffer from omitted variables misspecification, but not from ommited variable bias. This happens when the omitted regressor(s), which may be causally linked and useful in explaining the variability of the dependent variable, is/are mean-independent of the included regressor(s).  
In such a case, the correct specification is 
$$Y = b + aX + cZ + e$$
but still
$$E[b + aX + cZ + e\mid X] = b + aX + cE[Z\mid X] + E[e\mid X] = b + aX =  E[b + aX +  e\mid X]$$
since $E[Z\mid X]=0$, due to mean independence. This has the consequence that the estimator of the coefficient $a$ won't suffer from "omitted variable bias", but the estimated equation as a whole will be a sub-optimal predictor of the dependent variable.
So, the answer to the title question is "No", with the extension that the question "Is it possible for a regression to have the wrong functional form but not suffer from omitted variable bias?" is answered in the affirmative.
