Yes, we support SQLite on our Linux Managed servers. SQLite is pre-installed on our Linux Managed servers. Our PHP installation has support for SQLite 3.3 already included. We allow an unlimited number of SQLite databases.
If you want to use a newer version of SQLite, please open a ticket with the Done5 Crew at https://done5.com/clients/submitticket.php and we will see if we can install it for you.