I would first slightly rephrase the initial question:

Aucun magasin n’a en stock des aubergines et de la mâche.

You can’t skip the partitive article in such a list and mâche is more often used as uncountable.

The second proposal is the closer one but locating the pronoun before en stock is more idiomatic. In addition, the negative particle ne is expected in written or formal French:

Aucun magasin ne les a tous en stock. (tous referring to ces légumes).

I wouldn’t say aucun magasin ne les a toutes en stock, probably because légumes is masculine and you use a partitive article. The feminine would work with:

Les échalottes et la mâche dont on a besoin, il ne les a pas toutes en stock.

You might also say, to avoid the issue :

Aucun magasin n’a tout en stock.

Note: In the third link present in your question (wordreference: I have spoken to all of them), a sentence is reported to be incorrect although it is in fact much more usual and idiomatic than the only one marked as correct:

Je leur ai tous parlé. ✔? (rare)
Je leur ai parlé tous. ✘
Je leur ai parlé à tous. ✔ (usual)