Any “can we do X” question always has two aspects:
- Is it technically possible?
 - What about the business side of this?
 
On the business side, there’s an Apple part and a non-Apple part.  I presume that you’re concerned about App Review.  I don’t work for App Review and thus can’t offer definitive advice about their policies.  I suggest you review the App Review Guidelines.
The non-Apple part is the standard open source licence stuff, and I definitely can’t comment on that.
As to whether it’s technically possible, it’s generally fine to do this.  SQLite is just an open source library written in standard C targeting Posix-layer APIs.  You can generally add code like that to an iOS project without any problems.
There are, however, some potential caveats:
- Apple’s SQLite implementation has some Apple-specific tweaks.  These are not well documented, so it’s hard to say exactly whether this will be a problem for you or not.
 - You may run into problems if you try to combine code that uses Apple’s SQLite with code that uses your SQLite.  A good way to avoid any confusion is to static link your SQLite with its client code.
 
Share and Enjoy
—
Quinn “The Eskimo!” @ Developer Technical Support @ Apple
let myEmail = "eskimo" + "1" + "@" + "apple.com"