Unresolvable Ambiguity problem on Join Tables
GROUP BY ID. That's where the problem is. MySQL can't guess which table this column ID belongs to. There is an ID column in both the added wp_teleapo_supplier table and in one of the 3 others (I guess the wp_posts one).

All your column references should be prefixed with the table name (or alias): (object_id, term_id, post_type, post_title, post_content, ID). The only reason that these other references did not produce this error is because you were lucky and they only appeared in a single table, from the 4 tables in the FROM clause.

(some of the prefixes may be wrong. But if they are, you'll get an error and you can correct them:)

SELECT wpp.ID AS theid, s.ID AS supplierid, wpp.post_title, t.name AS category 
FROM wp_posts AS wpp
  LEFT JOIN wp_term_relationships AS r ON wpp.ID = r.object_id
  LEFT JOIN wp_terms AS t ON term_taxonomy_id = t.term_id
  LEFT JOIN wp_teleapo_supplier AS s ON wpp.post_author = s.ID
WHERE wpp.post_type = 'post'
  AND (wpp.post_title LIKE '%bag%'
    OR t.name LIKE '%bag%'
    OR wpp.post_content LIKE '%bag%')
GROUP BY wpp.ID ;