Since the effect occurs most intensely after taking the Vitamin C, my guess is that what you are experiencing is vasodilation and increased perfusion to the skin.
What does that mean? Your circulation to your skin improves after vitamin C and the spider veins (which are caused by valvular insufficiency within the veins themselves and weak vein walls) become more visible because they have more blood flowing through them.
Should you stop taking the Vitamin C? I wouldn't. If your skin is getting better circulation then it stands to reason that many of your other organs are as well and this is partly what we are seeking after all. Continued intake of Vitamin C and proline may actually help repair the valves, and strengthen the vein walls so that they no longer
over-dilate and bulge. Like with many natural treatments, often a condition seems to get worse before it gets better.
If the spider veins on your face are troubling to you in the meantime then you can try applying hemorroid cream to them. This is the same trick models use to get rid of dark circles under their eyes.
All in all, I would take this symptom as a welcomed sign of improved circulation. I think your progress would be accelerated at higher doses of Vitamin C, lysine and proline. CoQ10, grapeseed extract, ginkgo biloba, bilberry extract, and horse chestnut seed extract may also be valuable additions to your supplement regimen.
Suggested Reading:
http://www.emedicine.com/derm/topic475.htm
http://www.diagnose-me.com/cond/C329708.html
http://www.holistic-online.com/Remedies/varicose.htm
http://www.health911.com/remedies/rem_vari.htm
http://www.herbnet.com/ask%20the%20herbalist/asktheherbalist_questions%20on%20varicose%20veins.htm
http://www.bodyandfitness.com/Information/Womenhealth/varicose.htm
http://www.mothernature.com/Library/Bookshelf/Books/23/136.cfm
http://www.americanpregnancy.org/pregnancyhealth/skinchanges.html