Is there any reason why halogenation of substituted alkanes wouldn't occur through free radical substitution?
In principle, this reaction should be possible. However, the intermediately formed 2-hydroxy-2-propyl radical is not stabilized by the OH substituent, as the alternative mesomeric structure with C=OH bond and unpaired electron at O would violate the octet rule. 2-bromo-2-propanol should still be the major product, but there are also a lot of possible side products like multiply brominated compounds. Other substituents which stabilize an adjacent radical (by conjugation, for example) will increase the selectivity of the reaction. The possibility of reaction of the bromine radicals with the functional groups and sensitivity of the substrate to the reaction conditions (UV radiation etc.) also need to be considered. For the selective halogenation of certain substrates, often more suitable reagents than $\ce{Br2}/ h\nu$ are available, for example, NBS/AIBN for allylic bromination.