We claim the answer is $25$. To show that $25$ is achievable, take the set $S = \{p^2 : p \text{ prime, } p < 100\}$. There are $25$ primes less than $100$, so $|S| = 25$, and it is clear that its elements are composite, in the desired range, and pairwise relatively prime. To see that nothing greater works, observe that if $n$ is composite and its smallest prime factor is $p > 100$, then $n \geq p^2 > 10000$. Thus, if $S$ is a set with the desired property, then the smallest prime factor of each element of $S$ is less than $100$. However, all elements of $S$ must have distinct smallest prime factors, so $S$ has at most $25$ elements.