First of all, スクリーンショットを絵文字のし方 is ungrammatical. し方 works as a normal noun, rather than a verb phrase followed by the noun 方, and therefore doesn’t take an object with を like that.
スクリーンショットの絵文字のし方 is a grammatical noun phrase, where the noun し方 has two modifiers, both ending with の. However, its meaning is ambiguous, and therefore, you should avoid this structure and consider modifying a noun with a verb phrase, like スクリーンショットを絵文字にする方法, instead.
In general, a modifier with の seems to sound unambiguous and natural if the original particle is either (1) が to indicate the subject of the verb, (2) を to indicate the direct object of the verb, or (3) に to indicate a destination with a motion verb. As long as these conditions are satisfied, a noun phrase can have more than one modifier with の.
For example, the following sound completely natural.
彼女の字の書き方 (from 彼女が字を書く)
彼の自転車の乗り方 (from 彼が自転車に乗る)
These are precisely the cases when the meaning of the original sentence is understood even if the particles are omitted. While whether or not 乗る should be categorized as a “motion verb” may be debatable, 「彼、自転車乗る」 is definitely understood with no ambiguity.
Omitting に doesn’t always work, however. For example, 「スクリーンショット、絵文字する」 sounds broken, and this is probably why スクリーンショットの絵文字のし方 sounds unnatural. In another example, 「彼女、プレゼントあげる」 would be most likely understood as a colloquial form of 彼女がプレゼントをあげる, rather than 彼女にプレゼントをあげる, and therefore, this に should not be omitted if the latter is meant. Therefore, 彼女 is the giver in 彼女のプレゼントのあげ方, not the recipient.
By the way, this に could be converted into a modifier with への. For example, 彼女へのプレゼントのあげ方 sounds natural. In the case of your example, the verb should be changed from する to another with a clearer sense of “change”, as in スクリーンショットの絵文字への変え方.