Let’s flip this question on its head. Is there any reason to fix cells when staining for only surface markers? Short answer, no. Two reasons, one technical, the other practical.
Technical reason. Fixation typically involves para-formaldehyde. In the process, surface molecules are literally entombed/caged by formaldehyde molecules which do this by cross-linking them, i.e., fixing them. However, fixation can also destroy antigenic epitopes, i.e., those pieces of a cell-surface molecule that would be bound by an antibody specific for that molecule. An example from the time I used to work with mouse cells is the cell-surface molecule CD40L/CD154. Same monoclonal antibody (mAb) worked to bind it, i.e., stain it in FACS (Fluorescence activated cell sorting), when used before but not after cells were fixed. Reason? Anti-CD154 antibody’s epitope was destroyed by fixing. Another famous example is for the mouse CD4 molecule. GK1.5, RM4-4 and RM4-5 are among the most popular anti-mouse CD4 mAb clones sold by most vendors. Only experienced researchers would be aware that RM4-4 and RM4-5 work on both unfixed and fixed cells while GK1.5 doesn’t work on fixed cells. Again, reason’s the same. The formers’ epitopes are retained while the latter’s is destroyed by fixation. Whether a mAb’s surface epitope will survive fixation intact can only be determined empirically so decision to stain for it before or after fixation can also be done only after a case-by-case empirical test.
Practical reason. Fixing cells is a multi-step process that takes time and reagents. FACS for cell surface markers does not require fixing so why do extra, spending extra time and reagents when they’re unnecessary?