Does SQLite support replication?

Out of the box, no. There are a small number of third-party options:

SQLite sync: https://ampliapps.com/sqlite-sync/ This one seems attractive because it can replicate to other databases as well as SQLite, and doesn't modify the SQLite engine. I haven't tried it yet.

Litereplica: http://litereplica.io/ One way only. Seems to have been around a bit.

LiteSync: http://litesync.io/ Two-way replication. Pretty new, but an evolution of Litereplica so probably more mature than it appears. I have tried this a bit and it does seem to work smoothly, with a few bugs which the developer is looking at. You have to use the developer's modified SQLite engine, which seems like a concerning dependency. You also don't get much control, e.g. you can't say replicate now without reopening the database.

Brute force approach: Send it the ".dump" command to create a text representation of the data. Read that data in into the second database. Not sure you can use that.

If you need a fine grained update (sending a copy of each upto the other copy), have a look at sqlite3_update_hook

But how do you plan to handle errors? For example, what happens when the copy of the DB in app2 can't make an update for some reason?

To solve this, move the database to a server process and have the two apps talk to it.

Lsyncd - Live Syncing (Mirror) Daemon may be of use here. It uses rsync to do continuous replication on the file level.