Can a cointegrated variable be exogenous in first difference in a VEC model? If I have a variable C that is cointegrated with both variables A and B separately, can I use it in first-differenced form as an exogenous variable in a VEC model involving A and B?
 A: 
Can a cointegrated variable be exogenous in first difference in a VEC model?

and

[C]an I use it in first-differenced form as an exogenous variable in a VEC model involving A and B?

There is a difference between (1) a variable being exogenous and (2) justifying inclusion of that variable (or its first difference) on the right hand side of a model equation.
I will not go in the technical details of (1), but here is an illustration relevant for (2). You may have a data generating process such as 
\begin{aligned}
x_t &= x_{t-1} + u_t \\
y_t &= x_{t-1} + \Delta x_{t-1} + v_t \\
z_t &= x_{t-1} + w_t \\
\end{aligned}
with $(u_t,v_t,w_t)$ being i.i.d. random errors. Here $x_t$ is not affected by $y_t$ or $z_t$ and it does not adjust towards them. $x_t$ is only driven by its own error process $u_t$. Meanwhile, $y_t$ and $z_t$ adjust toward $x_t$ (with a lag of one period) and are driven by it. So then if you formulate a VECM for $(y_t,z_t)$, you have
\begin{aligned}
\Delta y_t &= (x_{t-1}-y_{t-1}) + \Delta x_{t-1} + v_t \\
\Delta z_t &= (x_{t-1}-z_{t-1}) + w_t \\
\end{aligned}
and obviously you should include $\Delta x_t$ in the equation for $\Delta y_t$ but not for $\Delta z_t$. (You can trivially modify the data generating process to include $\Delta x_{t-1}$ in the equation for $z_t$ or exclude it from the equation for $y_t$, and its role in the VECM would change accordingly.)
So

can I use it in first-differenced form as an exogenous variable in a VEC model involving A and B?

The above example shows that doing this may or may not make sense depending on the data generating process. However, you will need to include $x_{t-1}$ in levels in the error correction term. I think it is impossible that it would not appear in at least one of the error correction terms for $\Delta y_t$ and $\Delta z_t$ if both $(x_t,y_t)$ and $(x_t,z_t)$ are cointegrated.
A: A simple VECM model looks like $$\Delta Y_{t}=\beta\Delta Y_{t-1}+\Pi Y_{t-1}+\varepsilon_{t}
 $$
what you are talking about sounds like a simple extension $$\Delta Y_{t}=\beta\Delta Y_{t-1}+\Gamma\Delta X_{t-1}+\Pi Y_{t-1}+\Phi X_{t-1}+\varepsilon_{t}
 $$
In the above model, the $Y$ variables in levels are cointegrated with $X$. Further, a change in $Y$ depends on changes in $X$. But it does not include any particular model for the $X$'s, so you could consider it exogenous. There is no reason you can't estimate the model parameters with OLS, or other techniques.
That being said, it may not be reasonable to assume that $X$ is exogenous. If $X$ is cointegrated with $Y$, then you may want to revisit the exogeneity assumption. You may as well include it in the VECM. 
In addition, user @hejseb provides has provided multiple references (*Section 4.6) that warn that estimating the above model could result in faulty inferences about the cointegrating vector. The author recommends testing the assumption of weak exogeneity of $X$ by estimating the full model. 
