How come we can't use the equivalence point equation of NV=NV in this problem?
My question is, since are looking at the equivalence point in a titration, why can't we use $N_\text{acid} \times \text{Volume}_\text{acid} = N_\text{base} \times \text{Volume}_\text{base}$, since equivalents of acids and bases are the same at the equivalence point? If we can, how would we do it?


First of all, I would suggest that you stick to the original method using moles and proper mole ratios.

Please forget about this normality equation because it is only historically important and rarely anyone uses this concept in 2020 for research or academic work. Note, industrial analytical calculations still use this equation for very routine work. Still if you want to know why this normality equation fails in your case is as follows.

0) The $N_aV_a = N_bV_b$ relation holds if and only if all the concentration units are in normality, where $a$ and $b$ can be acid-base or oxidizing-reducing agent pair. It will also work for precipitation titrations etc. Corollary: This equation fails in titration calculations when the units are in molarity.

(i) This is a redox titration not an acid-base titration.

(ii) Let us change the subscripts for the sake for indexing, $$N_\text{ox}V_\text{ox}=N_\text{red}V_\text{red}$$

You have $\pu{0.01 M}$ $\ce{KIO3}$, what will be its concentration in normality units?

For that you need to see how many electrons are gained in the half-reaction...

$\ce{KIO3}\longrightarrow\ce{I2}$; Balance the equation and see how many electrons are involved in the redox process. See the example here
Balancing this reaction

Multiply the number of electrons involved in the half reaction by the molar concentration to get normal concentration of potassium iodate.

Apply the above equation and you will get the correct result but in normality units i.e. thiosulfate concentration in normality units.