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 A2 Hosting Guru Crew at https://my.a2hosting.com and we will see if we can install it for you.