Should I install wood flooring under the closet door track?
Really the only concern is clearance for the door. I'd say it's almost always going to be a better result to remove the track and install the floor. It'll most likely look better, be easier to clean, and be easier if you ever have to replace the door/track (eg: if new track is narrower or wider). 



When you say "wood vinyl flooring" everyone seems to be assuming you meant laminate or hardwood flooring, but I suspect you actually mean vinyl flooring with a wood print, which might be either click-together style or come in a sheet:



This adds a minimal amount of height, and depending on the doors it might work or you might be able to adjust the doors (there are usually ways to adjust the positioning of rollers). If you're not sure, it would be worthwhile to "dry fit" part of the new floor in with the doors on the track before you fully commit to this.

If the door can't fit, you have a few options:


Leave a gap in the flooring, leaving the track in place 
See if it's possible to adjust the top part of the track to raise it up by the amount the flooring adds. I did something similar to this that might give you an idea.
Replace the door with a shorter one or without a track (only makes sense if you were thinking of replacing it anyway)