Well the solution starts with a forced explanation and a sample of what your exit points may be.
For starters explain that to use SQLite it is very important to know in which architecture we are compiling . Since compiling in x86 SQLite.Interop.dll is one and in x64 it is another.
Taking into account that in some machines as you say it works (like me) it cost me a little more but I discovered where the problem comes from.
The problem is that for sure on those machines you do not have the Microsoft Visual C ++ 2010 Redistributable Package (x86) installed
Blessed: link
I have installed the two packages (x64 and x86) in my development machine but in the computer I had the same problem that you just downloaded the Redistributable Package (x86) and stop having the same error window like yours.
In summary:
If I compile my project in x86 (for 32 bits) I have to take into account:
Transport the System.Data.SQLite from x86 to the same root directory of the application.
Transport the SQLite.Interop.dll from x86 to the same root directory of the application.
Have the Microsoft Visual C ++ 2010 Redistributable Package (x86) installed.