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 הרה 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 last of the above cited entries:
[gradational variants]: הרר isolate; הרה implant; ירה cast
As explained on page 295:
[A gradational variant] involves five special consonants: י, ו, א, נ, and ה. These consonants play a special role with respect to roots whose third consonant is identical with the second. [...] Each of the word families created by the above formula is inter-related [sic]. Their respective meanings show degrees of intensity or similar activity in other spheres.
Thus, the basic root, הרר, means to isolate, while הרה and ירה are ways of doing so, by implanting it somewhere or casting it away.