According to the Etymological Dictionary of Biblical Hebrew, page 110. All brackets containing words are my expansion of their abbreviations; brackets containing ellipsis are skipping the examples provided, as they are irrelevant to the question. Italics are from the quote itself.
ירא: fear; call to constant attention
[explanation/commentary]: 1: fearing; being anxious of enemies [...] 2: being awesome [...] 3: being aware of a presence [...] 4: directing [...]
And further down on the page:
ירה: cast; shoot
[explanation/commentary]: 1: casting towards a particular place [...] 2: shooting at a target [...] 3: overthrowing by proxy [...] 4: early pelting rain [...]
So according to R’ Hirsch, teaching is not from either of these roots. It’s actually from the root הרה, page 61. (Boldface is mine.)
הרה: implant and absorb seed
[explanation/commentary]: 1: conceiving; becoming pregnant [...] 2: teaching; implanting seeds of knowledge ([Exodus] 4:12 והוריתי אשר תדבר also [Genesis] 26:5, [Exodus] 26:36, [Psalms] 9:21) 3: parent [...] 4: source of instruction ([Genesis] 22:2 אל ארץ המוריה)
Thus, הוראה and להורות, according to R’ Hirsch are, in fact, not from ירה or ירא at all.
However, that wasn’t the question. The question was if ירה and ירא are connected, and the answer is yes. R’ Hirsch is big on this sort of thing, and the dictionary provides a list of related roots. From the first of the above cited entries:
[cognate meaning]: limit activity [...] ירה cast
As explained on pages 293-294, words that share letters from similar parts of the mouth are related, where the letters toward the left are more “intense” than the ones on the right:
Gutturals: ע -ה -א -ח
Palatals: ג -י -כ -ק
Dentals: ד -ת -ט -ל -נ
Labials: ו -ב -פ -מ
Sibilants: ר -ז -שׂ/ס -שׁ -צ
So while both ירה and ירא describe limiting activity (fear, casting away), ירה limits it less so than ירא. In order, the full group is ירה, then גרע, then כרע, then קרע, then ירח, then ירא, and finally, the most limiting, גרה.