Plein is one of these adjectives that can be located either before or after the noun and that change their meaning depending on it.

Before a noun, plein means something like "broad", "in the middle" while after, it means "filled".

Plein vide appears two times in the linked document. I understand the first one

le vide vidé de son vide c’est le vide
c’est le plein vide

to be a pun leveraging this ambiguity (adj+noun or noun+adj) while in the second one, plein is definitely an adjective:

en plein vide.

Compare:

Un jour plein: A whole day, a full day. (opposite: un jour vide)

En plein jour: In broad daylight. (opposite: en pleine nuit)

Regarding "full emptiness" vs "empty fullness", I would translate the first one by vide complet or vacuité complète and the second one by intégralité vide.

01 le vide vidé de son vide c’est le plein
02 le vide rempli de son vide c’est le vide
03 le vide rempli de son plein c’est le vide
03 le plein vidé de son plein c’est le plein
04 le plein vidé de son vide c’est le plein
05 le vide vidé de son plein c’est le vide
06 le plein rempli de son plein c’est le plein
07 le plein rempli de son vide c’est le vide
08 le vide rempli de son vide c’est le plein
09 le vide vidé de son plein c’est le plein
10 le plein rempli de son vide c’est le plein
11 le plein vidé de son vide c’est le vide
12 le vide rempli de son plein c’est le plein
13 le plein vidé de son plein c’est le vide
14 le plein rempli de son plein c’est le vide
15 le vide vidé de son vide c’est le vide
16 c’est le plein vide
17 le plein vide vidé de son plein vide
18 de son vide vide rempli et vidé
19 de son vide vide vidé de son plein
20 en plein vide

There is no poetic licence involved, but then the proposed term, artistic license, describes perfectly what is at hand; why this term rather than poetic license becomes clear if, as I do, one considers that this is not poetry but an art parallel to poetry mainly concerned with peculiarities of semantics, and at that, semantics narrowed down to its relevance to the French language only, if, coincidentally, not entirely so.

That artistic license has been indulged in in this piece of writing is readily seen.
"Le vide de quelque chose" is defined for very few things; one can say "le vide de l’espace" (ref.), but "le vide du vide" is not in the least idiomatic, and that is what is implicitly defined in the first line: "le vide vidé de son vide" implies the assertion "il existe un état qui correspond au vidage du vide du vide.". Such unidiomatic constructions can be produced ad nauseam: le bois du bois, la chaleur de la chaleur, le rouge du rouge,…

The translation of "c’est le plein vide" as "it is full emptiness" is correct, "full" meaning "perfect". This is so because the place of the adjective; in that context it has the meaning found below (TLFI); this definition is compatible because "emptiness" ("le vide") is an abstract concept.

C. − Antéposé ou postposé. Entier, complet, sans restriction

1. Le plus souvent antéposé
   a) [En parlant d’une chose abstr.]
   Avec la pleine clarté de l’évidence (Clemenceau, Vers réparation, 1899, p.77).
   Le droit plein à la vie par le travail (Jaurès, Ét. soc., 1901, p.219).
   SYNT. Plein effet; plein épanouissement; plein succès; plein tarif; de (son) plein gré; plein exercice d’un droit; pleine confiance; pleine conscience; pleine liberté; plein et entier.

• le vide vidé de son vide c’est le vide
  c’est le plein vide

All of this is merely play with semantics; "to empty a hammock of its blankets", "to empty a pouch of its content", "to empty a vessel of the air it contains", all make sense; however, neglecting all logical considerations and insinuating that nothing (emptiness) is something that can be treated as something material, the concept of emptying something empty is put forward, hence the notion of emptying it of its emptiness which follows, and of course that must yield something; after such a null operation, fortunately there is no change and what you get is what you began with, which is emptiness, wherefrom (15) "le vide vidé de son vide c’est le vide", in English, "emptiness emptied of its emptiness is (again) emptiness" (As far as I can see, all of this is rather void of substance, mere idle play with concepts and warping of ideas.). As this null operation on the emptiness is still pretended to be non null, there must come something out of it, something that goes further than emptiness, and that is—again, fortunately—quintessential emptiness, the utmost in emptiness, perfect emptiness, which is embodied in "le plein vide" or in other (French) words, "le vide complet"; to render this perfection, intentionally, use is made of an adjective which in its plainest meaning (this one has nothing to do with the idea of perfection) signifies "full" (the contrary of "empty"; how funny…); of course, there is no pith in that: perfect emptiness is rhetoric and nothing more than emptiness. We end thus on another note of perverted meanings that leads nowhere except insofar as it enlightens us on the shortcomings of language and on how one can abuse it.

• le plein vide vidé de son plein vide
  de son vide vide rempli et vidé
  de son vide vide vidé de son plein
  en plein vide

The last part where this noun phrase is found elicits no new treatment ("plein vide" remains "full emptiness" as understood above) except for the last occurrence that has two interpretations. The preceding word is not a determiner any more but the preposition "en"; the meaning of "plein" can therefore different from that previously used. Below is the definition (TLFi) for this new possible second meaning.

2. En plein + subst. Au milieu, au cœur d’un lieu, d’une matière, d’un moment du temps, d’un événement, d’une action.

• Je ne demande pas absolument que vous me fassiez cela en plein jour, en plein Paris (Borel, Champavert, 1833, p.194).
• Je serais entrée en pleine révolte (Sand, Hist. vie, t.2, 1855, p.250).
• Nous sommes en plein orage. Les volets battent (Giraudoux, Sodome, 1943, i, 2, p.51).
  SYNT. En plein champ, en plein ciel, en plein désert, en plein milieu; en pleine campagne, en pleine forêt, en pleine mer, en pleine nature, en pleine rue; en pleine ville; en plein été, en plein hiver, en plein midi, en plein nord; en plein XXe siècle; en pleine nuit, en plein soleil, en pleine lumière; en plein cœur, en plein visage; en pleine figure, en pleine poitrine; en plein vol; en plein drame; en pleine activité, en pleine bataille, en pleine crise, en pleine retraite; en plein essor; en pleine forme; en pleine connaissance de cause.

For instance
en pleine lumière              →      in full daylight
en plein hiver                     →      in the depth of winter
en plein jour                        →      in broad daylight
(percuté) en plein œil        →      (hit) slap in the eye

"En plein vide" can also be "in the heart of the void", in the heart of vacuum", "in the heart of emptiness"; it seems you could use "in full emptiness" again but there is no context to differentiate "full emptiness" as "perfect emptiness" from "full emptiness" as "heart of emptiness".

As I understand this concluding part, there is the possibility of one meaning yielding two slightly differing variants, but the lack of punctuation makes that solution very doubtful as a unique way of reading these lines.
"Le plein vide" in "Le plein vide vidé…" (17) is the beginning of a sequence of appositions to "le plein vide" in "c’est le plein vide" (16), and it lasts to the end of the piece: in normal syntax—let’s not say "prose", as this is not poetry but a linguistic curiosity—there should be a comma after "c’est le plein vide". The flow of ideas is as follows. (16) It is full empti