I get way too excited looking at cover art, so this is a fun one for me. If you flip through English-licensed light novels and manga, certain names will pop up a lot: Noizi Ito (you probably know her from 'Shakugan no Shana' and 'The Melancholy of Haruhi Suzumiya'), abec (famous around 'Sword Art Online' covers), and Yoshitaka Amano, whose dreamy, painterly work graced many novel and special-edition covers like 'Final Fantasy' tie-ins. Range Murata is another name I always check for — his retro-futuristic vibe is unmistakable on special releases.
Beyond those big hitters, there are illustrators who became staples in light novel circles: Juu Ayakura for 'Spice and Wolf' style warmth, Kouhaku Kuroboshi for the quietly detailed look you see on series like 'Kino's Journey', and VOFAN, who has a striking, collage-like aesthetic on some licensed novels. Manga creators themselves (CLAMP, Takeshi Obata, Yana Toboso) sometimes draw covers for special editions, too.
If you're hunting credits, I usually look at the book's colophon or publisher page — Yen Press, Vertical, Seven Seas, and the like list artist credits. Also check the endpapers and author/artist Twitter or Pixiv accounts; many illustrators announce or showcase their cover commissions there.