Hi Justine,
I had varicose veins done in 2011 and am running. Initially I had one leg done using laser treatment to seal the vein (not sure if this is the same as the closure you mentioned) - essentially it is heated to extremes and sticks to itself. Following that I had to wear a full stocking for a few weeks which I hated and didn't exercise. I was then able to restart running.
A few months later the other leg was done which involved small cuts and picking out the vein as it was twisted but not to the extent of needing groin cuts. I asked the surgeon about running as he finished the op and he said I could run the next day if I was pain free (I wasn't!) so again took a couple of weeks out.
At no point was I advised to give up running. I did ask if perhaps running had contributed to my 'genetic curse', but the consultant reckoned that if anything it should help as it keeps circulation healthy and maintains a good body weight.
Perhaps you'll get further info to the contrary as things progress. There are some more discussions on here that may help if you search for varicose veins.