Though I have not used DiO stain but have experience with other stains. U can do the staining prior to fixing provided the follow up PFA fixation is done in conditions (like in dark) that do not quench DiO fluorescence.
I use neuro- and regular DiI, rather than Vybrant DiI, and have no problems fixing the signal in place. However, I have only tried retrogradely labeled axons of cultured neurons, so there is a lot of label in the membrane. Most of my work has been in vivo labeling.
One problem I have found with DiO is that it's green emission color can be difficult to see in paraformaldehyde fixed tissue because it is close to the autofluorescence of the cells. If you can get a lot of label into your cell membranes, you can probably fix them with 2 % paraformaldehyde. You might have to play with time and temperature, so that the cells get fixed but endocytosis is blocked (unless you don't care about getting label inside the cells). I would still avoid membrane-disrupting treatments, such as Triton X-100, or alcohols.
Molecular Probes makes a huge number of carbocyanine dyes, some of which have better retention after fixation. You should check with them, since I am not sure that the DiO comes in the more fixable form. However, the material linked by Christine references a Vybrant CM-DiI, so that may be an option if you need to label cells and then fix them.