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Ramba"m states that there is a mitzvah for every individual to write his own Torah. This mitzvah can be accomplished by writing one letter.
Why is there no mitvah for writing a mezuza. It seems that the Torah commands that we should write our own mezuza, as it says:
"You shall write them on the mezuzot of your houses and gates" (Devarim 6:9)
It seems that this is a commandment for every individual to write a mezuzah!
נפל האסימון - It hit me:
All Torah mitzvoth that have the word לכם mean that one must do it for himself.
Regarding a Sefer Torah Ramba"m Hilchot Tefillin Mezuzot veTorah 7:1 says:
מצות עשה על כל איש ואיש מישראל לכתוב ספר תורה לעצמו שנאמר ועתה כתבו לכם את השירה כלומר כתבו לכם תורה שיש בה שירה זו לפי שאין כותבין את התורה פרשיות פרשיות
It is a positive commandment for every Jew to write a Torah for himself, as it says, in Devarim 31:19 says "write for yourselves this song". This means that you should write for yourselves Torah that has this song, for we don't write a Torah in individual sections.
Excerpt from אדמת יהודה commentary:
מפסוק כתבו לכם משמע כל אחד ואחד דלכם כתיב כדאשכחן בארבע מינים שבלולב דאינו יוצא בלולבו של חבירו מדכתיב ולקחתם לכם
From this verse that says "Write for yourselves", we learn that it is obligatory for everyone to write for himself. We see similar language regarding lulav which also says "take for yourself" and thus, one does not fulfill the mitzvah by using someone else's lulav.
Similarly, counting of the Omer also says "for yourself". Thus, each person must count the Omer and one cannot count for someone else.
The commandment for mezuzah does not state the word lachem. Thus, there is no mitzvah for each person to individually write a mezuzah.
Ramba"m does state that one fulfills the Mitzvah of writing the Torah by having someone else do it. That's a separate discussion as to why that works, but it is not the "ideal" standard indicated by the Torah. G-d willing, I will edit in some information as to why that works. It's among some of the commentaries. Others are welcome to edit an explanation of that aspect into my answer.
