Can a curve's unit normal vector be determined using the second derivative? Because $$T = \frac {r'}{|r'|}$$ I was wondering whether or not it was also valid to solve for the unit normal vector with the second derivative without first solving for T: $$N = \frac {r''}{|r''|}$$

No, it's not. The  problem is that if you differentiate $T$, which leads to a multiple of $N$, you have to use the quotient rule, not just differentiate the top and bottom. You might want to try this with a simple curve like
$
( x(t), y(t) ) = (t, t^2)
$
at $t = 1$ to see that it doesn't work. 
