Explanation for the error
The immediate cause for the error message is that any explicit JOIN binds stronger than a comma (,) which is otherwise equivalent to a CROSS JOIN, but (per documentation):
Note: This latter equivalence does not hold exactly when more than two
tables appear, because JOIN binds more tightly than comma. For example
FROM T1 CROSS JOIN T2 INNER JOIN T3 ON condition is not the same as
FROM T1, T2 INNER JOIN T3 ON condition because the condition can
reference T1 in the first case but not the second.
Bold emphasis mine.
This is the cause of your error. You could fix it:
FROM  appointment_intakes
CROSS JOIN LATERAL jsonb_object_keys(data #> '{products}') keys
INNER JOIN appointment_intake_users ON ...
But that was not the only problem. Keep reading.
One might argue that Postgres should see that LATERAL only makes sense in connection with the table to the left. But that's not so.
Assumption
I added table aliases, and table-qualified all column names as suspected. While being at it, I simplified the JSON references and trimmed some noise.
This query is still incorrect:
SELECT i.data ->> 'id'          AS id,
       i.data ->> 'name'        AS name,
       i.data ->> 'curator'     AS curator,
       i.data ->  '$isValid'    AS "$isValid",
       i.data ->  'customer'    AS customer,
       i.data ->  '$createdTS'  AS "$createdTS",
       i.data ->  '$updatedTS'  AS "$updatedTS",
       i.data ->  '$isComplete' AS "$isComplete",
       count(k.keys)::numeric   AS "numProducts",
       u.created_at
FROM   appointment_intakes i
     , jsonb_object_keys(i.data -> 'products') AS k(keys)
JOIN   appointment_intake_users u ON u.appointment_intake_id = i.id
#{where_clause}
GROUP  BY i.id
Raw query
Based on the above and some more assumptions, the solution could be to do the count in a subquery:
SELECT i.data ->> 'id'          AS id,
       i.data ->> 'name'        AS name,
       i.data ->> 'curator'     AS curator,
       i.data ->  '$isValid'    AS "$isValid",
       i.data ->  'customer'    AS customer,
       i.data ->  '$createdTS'  AS "$createdTS",
       i.data ->  '$updatedTS'  AS "$updatedTS",
       i.data ->  '$isComplete' AS "$isComplete",
       (SELECT count(*)::numeric
        FROM   jsonb_object_keys(i.data -> 'products')) AS "numProducts",
       min(u.created_at)        AS created_at
FROM   appointment_intakes i
JOIN   appointment_intake_users u ON u.appointment_intake_id = i.id
--     #{where_clause}
GROUP  BY i.id;
Since you only need the count, I converted your LATERAL join into a correlated subquery, thereby avoiding the various problems arising from multiple 1:n joins combined. More:
You need to escape identifiers properly, use a prepared statement and pass values as values. Don't concatenate values into the query string. That's an invitation for random errors or SQL injection attacks. Recent example for PHP: