You have multiple problems.
Problem 1: Syntax error
Your query has a typo (a missing space). Your query code
$query = "INSERT INTO email_list (first_name, last_name, email)" .
"VALUES ('$first_name', '$last_name', '$email')";
produces this query:
INSERT INTO email_list (first_name, last_name, email)VALUES ('$first_name', '$last_name', '$email')
-- ^ syntax error, missing space
To fix it, change your code to this:
$query = "INSERT INTO email_list (first_name, last_name, email) " .
"VALUES ('$first_name', '$last_name', '$email')";
At least for testing purposes, you probably should look at the output of mysqli_error() instead of using a generic message like Error querying database. Even in production, you'll want to trap and log the real error somehow.
Problem 2: You don't select a database
Edit: I missed this in my first glance at your question, but as Stephen Cioffi points out, you also need to select a database before running your query. You can do this with the schema parameter to mysqli_connect() or by using mysqli_db_select().
Both of these issues—the typo and the failure to select a database—will cause problems; you must fix both.
Problem 3: Huge SQL Injection Vulnerability
This is not strictly part of the answer, but it's important. You are wide open to SQL injection. You need to use prepared statements. Otherwise, you are going to get hacked. Imagine that the POSTed firstname is this:
', (SELECT CONCAT(username, ',', password) FROM users WHERE is_admin = 1), '[email protected]') --
Your query becomes (with some added formatting):
INSERT INTO email_list (first_name, last_name, email)
VALUES ('',
(SELECT CONCAT(username, ',', password) FROM users WHERE is_admin = 1),
'[email protected]'
) -- ', 'value of lastname', 'value of email')
Then, when you email your users, somebody's going to get an email with a recipient like
"Duke,mySup3rP@ssw0rd!" <[email protected]>
And... you're hosed.
(Hopefully, you're salting and hashing passwords, but still, this is disastrous.) You must use prepared statements.