Is there an explicit mention mention of the Oral Torah in the Chumash? If not, why not?
There are a few hints to the Oral Torah in the Chumash, but nothing explicit.
One of these is found in Devarim 12:21:
וְזָבַחְתָּ מִבְּקָרְךָ וּמִצֹּאנְךָ אֲשֶׁר נָתַן יְהֹוָה לְךָ כַּאֲשֶׁר צִוִּיתִךָ
you may slaughter of your cattle and of your sheep, which the Lord has given you, as I have commanded you
The problem is that we don't find any command of how to slaughter in the Torah. So the Talmud writes (Chulin 28a):
דתניא רבי אומר וזבחת כאשר צויתיך מלמד שנצטוה משה על הושט ועל הקנה ועל רוב אחד בעוף ועל רוב שנים בבהמה
As was taught in a braysa, Rebbi (R Yehudah HaNassi) says, "וזבחת כאשר צויתיך" teaches that Moshe was taught (orally) on the esophagus and trachea (that they need to be cut so that the slaughter is kosher), and on רוב אחד בעוף (that most of one of the above need to be cut for the slaughter of birds to be kosher) and on רוב שנים בבהמה (that most of the food pipe and most of the wind pipe need to be cut for the slaughter of animals to be kosher)
I would like to supplement the important answers which precede mine:
The "Oral Torah" is mentioned "explicitly" in the sense that [according to our tradition] it is the subject being directly referred to in places such as the verse cited in DanF's answer. On the other hand, the "concept" or "doctrine" of Oral Torah is not stated explicitly in the same manner as many other concepts are only mentioned obliquely (resurrection of the dead, afterlife) or without full explanation ("angels") in the written Torah.
Yet despite not being "explicitly" taught, the written Torah is not self-sufficient for the implementation of its commandments and a number of important mitzvos implicitly demand recourse to an outside "living" body of knowledge. Notably, while we can learn certain things about the Torah's calendar from the written Torah we could not carry it out on the basis of the written Torah alone. Ironically some communities who rejected the Oral Torah objected to the "Rabbinic" move away from establishing the new moon by witnesses, a practice known from the Oral Torah but near as I can tell not explicitly mentioned in the Torah.
Similarly it would be impossible to identify which day was the seventh day for the observance of the Sabbath based off of the written Torah. Nor does the written Torah give explicit guidance on what constitutes a melachah ("work) on the Sabbath despite decreeing the ultimate penalty for Sabbath desecration.
In our own day we see the need for an outside body of knowledge (and what happens when it is lost) regarding the issue of techeles, while others might argue that any source for the "color" would be fine according to the written Torah it is not at all clear what color we would be looking for without resort to outside information to explain to us what color "techeles" is, never mind its source or how to process it.
Then there are other areas where we see apparent allusions, Shokhet has mentioned that the Torah alludes to a command regarding ritual slaughter that isn't discussed otherwise in the written Torah. How to to produce tefillin (without a reason we should believe that it wasn't a literal command). Perhaps most significantly the written Torah establishes a court of seventy elders with broad authority but without much explicit discussion otherwise.
Why G-d opted for the written Torah to require external information in order to observe the mitzvos may require a number of answers, but that it does is a matter of fact.
"... I shall give you the stone tablets, the Torah and the mitzvah that I have written to teach them".
Ramba"m explains that the word "Torah" means the written law, and the word "mitzvah" means the Oral Law.
Yes, this requires exegetical methods to determine the method of how he inferred this. So, when you say in your question, "explicit" meaning, "Does the Torah say the actual words, "Oral Law", the answer is, no. But, according to Ramba"m, the Torah does explicitly mention this. (We may get into semantics of what the term "explicit" means, in this case...)
I can't say as to why the specific words "Oral Law" are not mentioned. Perhaps, at the time that the Torah was written, which according to many beliefs (I think Ramba"m included) that it preceded the creation of the world, maybe when G-d created man, there was an initial expectation that man's level of understanding would be so high that he would be able to immediately understand all the nuances of the Torah as written without further translation?
Another possibility - When I first learned Chumash in 2nd grade, we were told to translate the Hebrew words, literally, without looking in Rash"i or any other commentaries. (Yes, I had a very high-functioning demanding yeshiva. I wish others would model this example, today.) The teacher wanted you to really toil in the Torah. Rashi was like the "Monarch Notes". In a similar sense, perhaps, God wants people to toil and work over the literal translation a bit, and think it through prior to getting the explanation? Again, conjecture on both of my answers...