I wondered just how difficult it would be after I replied and lo, it's actually pretty easy to get an SQLite connection from iLogic.
All you or any future reader needs to do is copy the relevant files to the iLogicAdd folder:

The version of the SQLite files needs to be the x64 build, but with the files in-place as per the screenshot above, you can simply do this:
Option explicit On
AddReference "System.Data.SQLite.dll"
AddReference "System.Data"
Imports System.Data
Imports System.Data.SQLite
Dim sqlite_conn As SQLiteConnection
' create a new database connection:
sqlite_conn = New SQLiteConnection("Data Source=F:\path\to\your\database\file\iLogic Test.db;Version=3;")
sqlite_conn.Open()
Dim sqlite_cmd As SQLiteCommand = sqlite_conn.CreateCommand()
sqlite_cmd.CommandText = "CREATE TABLE test (id integer primary key, text varchar(100));"
sqlite_cmd.ExecuteNonQuery()
🙂
PS. I have attached the sqlite database I tried to attach the sqlite database I created to test this rule, but the forum said no, so if you'd like a copy please let me know! FWIW: I used the sqlitebrowser which I installed using chocolatey from here.