Can special relativity distort the relative order in which events occur?

In special relativity, you think of a 4-dimensional space-time. The key point here is that two events, 1, and 2 happening at $t_{1}, x_{1}, y_{1}, z_{1}$ and $t_{2},x_{2},y_{2},z_{2}$ have a distance given by ${}^{1}$

$$(\Delta s)^{2} = -c^{2}(\Delta t)^{2} + (\Delta x)^{2} + (\Delta y)^{2} + (\Delta z)^{2}$$

Now, we can therefore give any two events a unique relation to each other:

1) $(\Delta s)^{2} < 0$: These events are considered timelike separated.

2) $(\Delta s)^{2} = 0$: The events are lightlike separated

3) $(\Delta s)^{2} > 0$: The events are Spacelike separated

The key point is that, in all reference frames, timelike and null events happen in the same order (this is derivable from the fact that all observers follow timelike paths).

But, you can also show that there is no unique ordering of spacelike events--events that happen in order 1 > 2 > 3 in frame A may happen in order 2 > 1 > 3 in frame B. In fact, for any two spacelike separated events, it is possible to find a reference frame where you can reverse the order in which they happen.

So, the answer to your question is "maybe", but the trick would have to be that you throw the second dart much more quickly than the first, and it hits the dartboard a distance X a way at a time T later than the first, in such a way that $T < X/c$. You do this, then there will be a reference frame where it appears that the second dart hits first.

${}^{1}$here $\Delta t = t_{2} - t_{1}$, etc

No, special relativity does not violate the order in which events take place. Causality is preserved, two observers in different frames of reference will always agree on which dart hits the board first, as the movement of a dart corresponds to a timelike curve in four-dimensional spacetime. The only quantity they will disagree on regarding the impact is the difference of the two times, this can be explained in terms of time-dilation.