Can I reprocess vinegar hot pepper jelly?

Yes, though to be clear, you unseal, empty the jars into a pot, heat & add sugar, (while re-cleaning/sterilizing the jars) then fill the hot jars and process. You don't just add sugar to the jars.

To suit the food safety fanatics, use new lids.

I, personally, figure that if I use old lids and they seal, it's fine, because it sealed, and I know that because the lid is designed to indicate that. Sealed is Sealed.

If I use old lids and they don't seal, I know they didn't seal, and treat appropriately, just as I would new lids that didn't seal. But the official recommendation is to toss the lids and use new ones every time, even if they were new until last night and have only been on the jars for a day (unless you are using "specifically made to be reusable lids.") I note that the official doom and gloom warns of failure to seal (which is obvious) not some dire subtle thing, so I choose to ignore them on this one point so long as a lid is in good physical shape. But it's not the official recommendation.