Calculate the probability of a variable on a Bayesian Network

Let's assume we have a bayesian nework with discrete variables like shown below. For simplicity, assume all variables are binary.

Assume we know the states for V1 and V2 and we want to calculate the probabilities for V3 being in either of its states.

Is it sufficient to just observe the values in V3's CPT, given that we know the state of its parents or do we have to calculate the joint probabilities over all network variables?

It depends on what probability you want. If you want the marginal $$P(V_3=v_3)$$ then observing only realizations of $$V_3$$ is enough. However, your goal seems to be
given the states for $$V_1$$ and $$V_2$$, what is the probability for $$V_3$$ taking a specific value $$v_3$$.
In Bayesian Networks, one usually computes the kernels $$P(V_i\mid \mathrm{Pa}(V_i))$$ where $$\mathrm{Pa}(V_i)$$ are the parents of the node $$V_i$$. In this case, you need to observe the variable $$V_3$$ jointly with its parents $$\mathrm{Pa}(V_3) = \{V_1, V_2\}$$. This is because in a DAG the local Markov condition allows for the factorization: $$P(V_1, \dots, V_n) = \prod_{i=1}^nP(V_i\mid \mathrm{Pa}(V_i))$$ So it is enough to observe $$V_1, V_2, V_3$$ because of this factorization. You do not need to condition on the descendants or non-descendants of $$V_3$$. In your case the factorization becomes $$P(V_1,V_2,V_3, V_4, V_5) = P(V_1)P(V_2)P(V_3\mid V_1, V_2) P(V_4 \mid V_3) P(V_5\mid V_3)$$