You probably do not need to switch yet. SQLite allows many readers but only one writer at a time, and Python's sqlite3 waits only five seconds by default for the other writer before giving up. Two changes cover most cases:
import sqlite3
con = sqlite3.connect("app.db", timeout=30) # wait up to 30 seconds for the other writer
con.execute("PRAGMA journal_mode=WAL") # readers no longer block the writer
The longer timeout makes the second script wait its turn instead of failing. WAL mode (write-ahead logging) lets readers and one writer work together; it is stored in the database file, so you set it once.
Also keep each write transaction short: open it, write, commit(), and do not hold it open while the script does slow work such as calling an API. If two scripts regularly need to write for a long time at once, that is the point to consider PostgreSQL.