A bunch of examples:
(1) 脚不小心踢到椅子上面[src]
(2) 在上课，后面的人不小心踢到我椅子了[src]
(3) 孩子不小心踢到椅子[src]
(4) 贾玲跳舞,不小心踢到椅子,椅子腿断了,陈赫非常关心![src]
(5) 脚肿了，是不小心踢到椅子脚上，小脚趾肿了[src]
I don't understand why there's no 地 here, i.e., 不小心地踢, as per
Adj. + 地 + Verb
地 (de) Before Verbs and Turning adjectives into adverbs, Chinese Grammar Wiki
Maybe there's an exception here.
Question: Why is there no 地 in 不小心踢到椅子?
Strictly speaking, your translation is incorrect and that is what is tripping you up.
不小心 has many interpretations here, and the one employed in all these examples is along the lines of "wasn't paying attention" or "was not being careful" -- this implies it is NOT actually playing the role of an adverb in the sentence. As such, 地 is not required. In essence, it is two subsequent actions.
for example:
他不小心摔倒了 -- He wasn't paying attention, and fell down.
compared to
他不小心地摔倒了 -- He carelessly fell down.
The spirit of the sentences is basically the same, but your question appears to be more about the nitty gritty details of what role 不小心 is playing in the sentence. If you wish to translate it as an adverb, then you are right, there should be a 地 there, but that's not the only way to translate 不小心. According to youdao, it can play a noun, adjective and adverb.
It will make more sense to you if you understand it as:
孩子[由于不小心]踢到椅子.
不小心 here is the cause of 踢到椅子. In English, it can be The kid kicked the chair due to his carelessness.
