It isn't too uncommon for Captain America's shield to have the paint chipped off or damaged.
However, it's soon back to normal, every time. Who repaints it for him when this happens?
Here we see the good Captain himself, replacing a broken leather strap and touching up the paint on his shield (note the can of red paint - it actually looks like regular house paint - on the work bench).
Unfortunately for the Cap, shortly after he finished this repair work, the shield was stolen by a villain named Batroc, stripped of all its paint, and analyzed in the hopes of reproducing it to create invulnerable armor for the baddies. We can assume that this meant Cap had to paint the damned thing twice in a week.
The Cap obviously wants his shield back, so he raids the baddies' HQ, and finds a room full of unpainted, apparently identical shields; he basically has to figure out which one is his by throwing them all at people and seeing which one doesn't suck at flying.
By the end of the story arc, Cap and his beloved shield are back together, paint and all.1
In another story arc, the shield has been shattered, and Tony Stark is kind enough to take a trip to Svartalfheim to have it repaired by the Dwarves. It ends up stronger than ever, but signs of the damage are still visible. There isn't any mention of painting, but Tony does offer to "buff" it for the Cap.
We might assume that if Tony is willing to polish the shield for the Cap, he would probably agree to repaint it, too - or at least pay a lackey to do it for him.
In this issue, Cap is brainwashed by Nazis. After painting a swastika on his shield, the baddies send Nazi Cap out into the world to do Nazi stuff.
Daredevil finds out about this and fights the Cap, eventually tricking him into getting oil on his Nazi shield.
The oil mixes with the Nazis' paint, the paint dissolves, and the shield we all know and love is revealed, unharmed by either the paint or the oil.
Apparently, the red, white, and blue isn't harmed by the paint and oil because this iteration of the shield was colored while it was still being forged. It isn't paint at all - it is the color of the metal itself.
We don't know who paints the shield when it needs to be repaired, and we don't even know who, exactly, painted it the first time. That having been said, it seems safe to assume that the initial paint job was done by someone who worked for Howard Stark.
When Steve first finds the shield in one of Howard Stark's many piles of incredible, groundbreaking technology that he invented and then promptly forgot about, it is bare metal:
Steve asks what it is, and Howard says "Oh, that thing sucks. I mean, it's totally indestructible, and a god could literally smash it with a magical hammer without denting it, but who cares?" Steve decides he wants it anyway - it is much cooler than the kite-shaped shield he'd been carrying thus far - and Howard obliges, because as far as Howard is concerned, the shield is basically garbage.
The next time we see Cap with his new shield, it has the trademark red, white, and blue. Again, we don't know who painted it, but the obvious implication is that someone at Stark Industries did it.
Stark Industries made the shield, and almost certainly applied the first paint job, and the Cap now works with the heir of the Stark corporation, so it stands to reason that the Cap could get the shield touched up free of charge whenever he likes - all he has to do is call Tony (or Pepper, which would probably be a better idea, since Tony loves to mess with the good Captain, and might screw around with the shield for a laugh) and ask.2
1 Although the Cap seems to be a competent handyman here, his repair work abilities are limited. Case in point: That time his shield was shattered and he patched it back together with... um... duct tape.
2 This line of reasoning - i.e., "the guys who gave the shield to the Cap in the first place would probably be willing to repair it for him later" - doesn't hold up in all cases. For instance, in one timeline, Captain America was presented with his shield by US President Franklin Delano Roosevelt.
Between pulling the US out of the Great Depression and engineering our entry into the biggest war in human history, I doubt FDR had the time or the inclination to concern himself with repainting Captain America's shield, and I can't imagine the Cap being oblivious enough to ask him to do so.