A table represents a relationship among values. (Some values identify entities.) A database can't be used until we are told what each table--base or query result--means: what business/application relationship its rows satisfy. (Ie, what is its (characteristic) predicate.) What are the relationships for your tables?
To query we express a relationship in terms of base relationships then express its table in terms of the corresponding base tables. Eg a join returns rows satisfying one relationship and another. So again, we need to know tables' relationships in business/application terms.
Cardinalities & constraints are properties of relationships given what situations can arise. They aren't needed to update or query. They can guide design & are used for integrity.
When, given a clinic, you talk about "its" suppliers, you do not say what you mean. "Has", "its", "for", "references", "appropriate"--all mean nothing--they refer to related entities, but they don't say how they are related in terms of the business/application. This design contains no explicit relationship on clinics & articles. If it did, you've said nothing by which we could put the right rows in or see the rows & know about the situation. Still, you could then derive clinic-supplier rows where the supplier is "for" some article a given clinic "has". But is that the relationship you mean?? Eg if you want pairs where the clinic is allowed to "have" the supplier "for" some articles, that's a new relationship/table that cannot be derived from what you have.
creating a "null" article for each supplier in article_supplier
That relationship/table is a certain combination of article_supplier & the relationship/table just described. But it is simpler to just have those two.
creating a foreign key in supplier that references the appropriate clinic
That would mean that if a supplier "has" more than one clinic then there can be rows in the new version that differ only by all the other columns; normalization theory says that's worse than the original design and a clinic_supplier relationship/table. And it means that if a supplier "has" no clinic you would need something like a nullable FK (foreign key).
So you likely want a clinic_supplier table. But you should post a new question in which you actually say what business/application relationships you are talking about.
Your question & the following are essentially duplicates in that the basic principles/notions to obviously apply to answer them are all the same:
How do I find relations between tables that are long-distance related?
Required to join 2 tables with their FKs in a 3rd table
Best Solution - Ternary or Binary Relationship
You need to read an information modeling & database design textbook.