Unfortunately, it is not possible to use SQLite under medium trust. It does PInvokes against a native dll and this is not allowed under medium trust.
Some hosts if you inquire will move your site to a server running in Full Trust, I know DiscountASP and Crystal Tech have both done that for me in the past, so you might ask your host about it.
I have heard rumors that maybe in .NET 4.0 SQLite will be installed in the GAC so that it could run under medium trust, but it is not a confirmed rumor.
I also have my eye on csharp-sqlite, which is a port of SQLite from C to C# and thus fully managed code so it "should" be possible to use it under medium trust, but I don't think it can be dropped in as a replacement for SQLite and not sure it will perform as well as the C version.
Hope it helps,