Can I use a function for a default value in MySql?

No, you can't.

However, you could easily create a trigger to do this, such as:

CREATE TRIGGER before_insert_app_users
  BEFORE INSERT ON app_users 
  FOR EACH ROW
  SET new.api_key = uuid();

As of mysql v8.0.13 it is possible to use an expression as a default value for a field:

The default value specified in a DEFAULT clause can be a literal constant or an expression. With one exception, enclose expression default values within parentheses to distinguish them from literal constant default values.

CREATE TABLE t1 (
  uuid_field     VARCHAR(32) DEFAULT (uuid()),
  binary_uuid    BINARY(16)  DEFAULT (UUID_TO_BIN(UUID()))
);