My supply function is $Q = a + bP + u$. My demand function is $Q = c + u$. The error terms $u$ in both equations are mutually i.i.d random variables with a mean of zero and a constant variance. My question is: since the demand curve is a constant, is price an endogenous or exogenous variable?
It is endogenous, because a demand shock will affect prices.
To see this more clearly, add another regressor to the demand, for instance, income. It is clear that income is exogenous, as it is not affected by prices or quantities, or shocks. This cannot be said of prices.
This is also evident if you write the system as it should have been written in the first place:
$$ P = \alpha + \beta Q + \epsilon $$ $$ Q = c \quad \quad \ + \mu$$
This is a simultaneous equation system, where both $Q$ and $P$ are endogenous. Here you can see more clearly that a demand shock $\mu$ affects prices via $\beta$ (if $\beta=0$, the two equations are independent).
This means that the OLS estimation of $b$ is inconsistent. A consistent estimation can be obtained with 2SLS, using a proper instrument.