来る and する are two irregular and most commonly used verbs. From my observations, 来る is spelled with kanji. Even though there is a kanji version of する 為る it is usually not used and I wonder why. Are there any cases where the kanji version of する is used in modern Japanese?
The reading 為る【する】 is not a reading contained in the jōyō kanji, which is yet another reason it is not commonly used. (You don't learn about it in school, you will have trouble publishing work with all instances of する replaced by 為る, etc.)
The why has to be speculation, but to me it makes sense to drop the kanji for the second-most used verb in the language; after いる, which derives from 居る, but is usually written in kana.
Of course, there is already the general convention to use only kana for subsidiary verbs, so that ～ている and also ～てくる will usually be written in kana.
Something deserving a mention on the topic of する and kanji is that there exists an ateji for the renyōkei (し) of the verb する, namely 仕, which appears in many common words such as
In compounds, which are usually formed with the renyōkei of a verb, the renyōkei of する is written in kanji. In that sense, one could say that する does have a commonly associated kanji, even if it is not used as *
する is "to do" and is essentially an auxiliary verb. Auxiliary words are generally not usually written in kanji. However, 来る can be a substantial verb meaning "to come" and is often written in kanji for this usage. When used as an auxiliary verb, though, it is usually written as くる and almost never as 来る.
Working in Japan I have never seen (or even heard) about the kanji version of する. Now that you mention it I would guess that the reason is practical.
For one, you use する so much that it would be tedious to write out the kanji every time.
Also, there are other words that sound like する like 擦る and 磨る and the non-kanji one is just a way to distinguish that word from the others.