This kotobank.jp link says that『日本国語大辞典』refers to a kanji version of「そそっかしい」as「麁相かしい」, and gives a quote from the novel (section) 式亭三馬『浮世風呂・2・下』:
私が一体麁相かしい性で
Here's a page from『浮世風呂』(taken from 人文学オーペンデータ共同利用センター) with that quote highlighted:

This is my transcription:

If you believe that this furigana says「[麁相]{そそっ}か」(refer to a hentaigana chart), and trust the authority of『日本国語大辞典』, then「そそっかしい」can be written as「麁相かしい」.
In the『日本国語大辞典』entry, the only quote including「そそっかしい」which has kanji is「麁相かしい」.
Additionally, according to this dictionary,「そそっかしい」is actually just a variant and is derived from「そそかしい」with the addition of the sokuon「っ」. Under「そそかしい」, more kanji variants have been recorded, including
- 『和英語林集成』(1867年初版): 「[粗粗]{そそ}かしい」
- 夏目漱石『明暗』: 「...彼を[忽卒]{そそか}しく」
Thanks to @By137 for providing this info
尾崎紅葉『金色夜叉』is another novel which possibly uses that reading:

My transcription:
[不相變麁相]{あひかはらずそ〻ツ}かしいね
In both cases, it looks like「ツ」(or its very close hentaigana equivalent「𛁪」, U+1B06A) is used for what is now「っ」, which may have been common orthographic practice at that time (these two were written around 1800-1900).
Of course, there might be other kanji representations of「そそっかしい」used around this era; many of these texts are not digitised (or properly checked after digitisation) so they remain unsearchable on the web.