It is possible to replace the LCD -- however for a proper official part, it would run you about $90 on average. Disassembly of any PSP unit is not easy, there are many things that can break with the slightest wrong movement. (Including the LCD connector!)
Your most economical bet is to try the eraser method (a brand-new pencil eraser, gently rub the scratch until it disappears) or a scratch repair kit purchased from a store. There are other methods involving things like rubbing alcohol, but if you get the mixture wrong, you'll damage your LCD far worse than than it already is.
Good luck!