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For instance：

昔の日本は身分の差が大きく厳しい上下関係があります。

Is 大きく modifying 厳しい? But the way I understand this sentence is in ancient Japan, there were large social divides and strict social hierarchy, which is in effect 大きくて厳しい上下関係があります。

Thank you!

Can adverbs modify adjectives?

Yes. That's part of what adverbs do. Consider:

• [と]●[て]●[も]●赤【あか】い車【くるま】
  
  a very red car


• [ま]●[ぶ]●[し]●[く]●明【あか】るい青空【あおぞら】
  
  a dazzlingly bright blue sky

Is 大きく modifying 厳しい?

No. At least, not directly. In your sample sentence, as you correctly parsed it, the 大【おお】きく is not being used as a direct modifier on the following adjective 厳【きび】しい, but is instead being used in a way that can be understood either as a conjunction, as you suggest with your 大【おお】きく[て]● alteration, or as a kind of adverbial clause that modifies the whole rest of the sentence.

One of the grammatical clues telling us that this 大【おお】きく is not just an adverbial modifier for 厳【きび】しい alone is that the 大【おお】きく has a subject marked with が immediately before it, telling us that the 大【おお】きく is in fact being used as the predicate of a descriptive phrase with that が-marked subject as the head noun of that phrase.

The grammar

This kind of adverbial ending to a phrase can also happen with verbs, where, by one analysis, the て is omitted. Example:

• ご飯【はん】を食【た】べテレビを見【み】る
  
  [I] eat and watch TV

This kind of construction joins clauses using the 連用形【れんようけい】: the -masu stem for verbs, the く adverbial for -i adjectives, the に adverbial for -na adjectives -- literally, the 形【けい】 or "form" that 連【れん】 or attaches to a 用言【ようげん】 or inflecting word.

How this affects the meaning

In terms of nuance, the meaning of the phrase ending in the 連用形 is a bit more closely linked to the meaning of the following phrase than if you use the ～て conjunction. Compare:

• まぶし[く]●明【あか】るい青空【あおぞら】
  
  a dazzlingly bright blue sky


• まぶし[く]●[て]●明【あか】るい青空【あおぞら】
  
  a dazzling and bright blue sky

Or:

• ご飯【はん】を食【た】[べ]●テレビを見【み】る
  
  [I] eat and watch TV (probably at the same time)
  


• ご飯【はん】を食【た】[べ]●[て]●テレビを見【み】る
  
  [I] eat and then watch TV (as distinct actions, possibly separated in time)
  

Looking specifically at your sample sentence:

• 身分の差が大き[く]●厳しい上下関係があります。
  
  There is a big difference in status, and [thus] a strict hierarchy.


• 身分の差が大き[く]●[て]●厳しい上下関係があります。
  
  There is a big difference in status, and [separately] a strict hierarchy.

I'm stretching the English translations here to try to emphasize the distinction in the Japanese, but hopefully this helps shed some light on why an author or speaker might use the 連用形【れんようけい】 to end a clause, as opposed to the ～て conjunctive form.