This is a good example of how kanji spellings can obscure word derivations. 😄
閉【と】ざす, despite the kanji, is etymologically unrelated to 閉【と】じる.
This first appears in a text from 810, and was conjugated as a 上二段活用動詞【かみにだんかつようどうし】 or "upper-bigrade conjugation verb". These are the precursors to your modern "vowel-stem verbs" where the stem (the part that stays the same across all conjugations) ends in //i//. The original 終止形【しゅうしけい】 ("terminal form", a.k.a. sentence-ending form, predicative form, plain form, dictionary form) was とづ.
The verb is one of the rare ambitransitive verbs in Japanese, meaning it can be used as both transitive and intransitive. It looks like the transitive senses are attested first (in 810), followed later by the intransitive senses starting in The Tale of Genji in about 1000.
The core meaning has been "to close".
This first appears in roughly 900, conjugated as a 四段活用動詞【よだんかつようどうし】 or "quadrigrade conjugation verb". These are the precursors to your modern "consonant-stem verbs" where the stem (the part that stays the same across all conjugations) ends in a consonant -- in this case, //s//.
This verb is also attested as ambitransitive, with the intransitive senses appearing from around 900, and the transitive much later, from around 1885. The intransitive senses appear to have become obsolete, likely due to the ～す ending giving an impression that this verb should be transitive.
The core meaning has been "to close and lock".
This aligns with the compound meaning indicated by many dictionaries I've consulted -- とざす from 戸【と】 ("door") + 鎖【さ】す ("to close", with overtones of "to lock", from the base sasu sense of "to pierce with something long and skinny", presumably in reference to either the bolt or key of a lock).
Considering the compound nature of とざす, it is reasonable to wonder if とじる is also a compound. Given older spelling とぢる and original base form とづ, this looks like it might be from 戸【と】 ("door") + 出【づ】 ("to come out"), perhaps from the way that doors in Asian architecture are often pocket doors that slide out from the wall.
However, 出【づ】 consistently conjugates as a 下二段活用動詞【しもにだんかつようどうし】 or "lower-bigrade conjugation verb" with conjugation stems ending in //e//. I cannot find any evidence of this ever appearing as an upper-bigrade verb instead, so this looks like a long-shot theory at best.
I often refer to the 日本国語大辞典【にほんこくごだいじてん】 or NKD. This dictionary often includes more historical information about words than most other dictionaries that I've used. One version of the NKD is made available for free by online reference aggregator Kotobank.