Where can I get more arrows in WOW?

Lol sure. I don’t think you can in the starting area. The next island will have a quest hub that will have a vendor to sell you ammo. You may also be able to track ammo vendors from the magnifying glass on your mini map.

Vendors is correct, after starting area. Although most starting areas usually have an ammo/junk vendor hanging around.
This, that and here are all you need for WoW. All I ever used anyway.

1) There are arrow vendor all big/small cities so just track ammo on your mini map
2) Use wowhead.com and not thottbot, it is so outdated and looks like a site from the 90’s

wooo vendors

The notbeinganoob store.

Just as an update for any players who might be more recently looking for help on this same thing:

As of the recent patches bows and guns no longer use ammo, so buying them is no longer possible since they aren’t necessary anymore.