SHA256 output to 0-99 number range? Is it mathematically possible to take a SHA256 hash and turn it into a 0-99 number where each number in 0-99 range is equally likely to be picked?

As a 256 bit hash means the highest value possible is 2^256, this is not a nice or "round" number for the purposes of simplifying a hash down to ranges more natural for humans.
If you mean exactly as likely, no, because the number of possible hashes is not a multiple of $100$.  This is assuming all the hashes are exactly equally likely.  You can come very close just by taking $SHA256 hash \pmod {100}$  This will be within one part in $\frac {2^{256}}{100}$, which is a very small number.  If you want truly equal, check that the hash is in the range $[0,100\lfloor \frac {2^{256}}{100}\rfloor)$ and take it $\pmod {100}$  If it is above that, hash something else and try again.  Your odds of failing the check are very small, and your odds of failing twice are very small.
In the general case: if one wants a result in the range $0..n\text{-}1$ then use $\bmod n$ and discard if the output  $&gt;n\lfloor  \frac {2^{256}}{n}\rfloor$ and try another input. This is called the discarding technique; it is commonly used to extract a number in a range from a (pseudo) random number generator.