Full disclosure here, I'm not Orthodox, I'm a Reform girl, but this is what I was able to grab, insomuch as understanding where the Orthodox position stands on rabbinic authority.
First, some clarification. Halacha is Jewish law with a practical basis. Aggadah, which was not passed down from Halacha L'Moshe Mi'Sinai, is Jewish folklore. Today, we won't be discussing anything to do with the aggadah, as it does not pertain to our study.
I. PROOFS FROM OUR TORAH
The first proof comes from Shemot 23:19. There, we find חלב, now, whether it's chalav or chelev is unclear if you don't have the Torah she-be-`al peh as a guide. Keep in mind that the vowels were invented in the 10th century, however, it's clear they were part of the mesora, and hence, passed down from one generation to the next. Other examples we can choose from have already been discussed, such as how one keeps Shabbos, and what does it mean to be punished by death if you refused to keep it.
I want to move on, specifically, to biblical references. One prime illustration comes from Nechemiah 13:15-17, where some Jews violate Shabbos by separating grain from the husks via a squeezing press. The Navi quoted these strict legal bindings from Torah, but none of the above are found in the Torah she-bi-khtav. Clearly, then, he meant the Oral Law.
As far as rabbinic authority goes, we learn from Vayikra 18:30 and Shemot 18:13-22 that legal safeguards are to be erected by the elders to deal with case law and hermeneutics - in short, Klal Yisrael .
This teaches us that the correct interpretation of passages such as Yehoshua 8:34-35 are to be understood in light of the above.
Note that while there were debates over a certain takkanah, there was never an issue between Chazal over the principle of there being an Oral Law in the first place. This means that there has always been a legal procedure in which Chazal could study and make rulings. Note further that the Gemara purposefully decided to record all opposing views so that when a future Sanhedrin is reestablished, they will know that these discussions have been properly dealt with.
II. HISTORICAL PROOFS
Even though verses such as Devarim 6:7 and Tehillim 1:2 exist, secular scholars still come away with the claim that our ancestors forgot the law for a thousand years (citing Melachim Bais 22:8). The verse in question, however, isn't talking about the founding (or invention) of the first Torah scroll, but rather, the discovery of the original parchment. Shmuel Alef 4:19-22 is clear how the people felt about it, and they sure weren't up in arms, as secular scholars would have us believe. Moreover, one may compare innumerable Torah mandate with its application elsewhere in Tanakh before the age of Yoshiyah HaMelech .
Regarding the Isiyim, they too were Perushim, though it is a little known fact. Evidence for this comes from the strict nature of their Shabbat observance, such as added time before nightfall; prohibiting business dealings, perfumes, labor, cooking; while permitting walking a thousand cubits and breaking Shabbos to save a life.
Now, if we turn our attention to Josephus, we'll find that he too followed the Oral Torah, and ascribed it all to Moshe Rabbeinu (see Antiquities Book 3, 5.6). To be brief, he agreed with the Mishnah's ruling that Nisan was the first of the month, and that the months were to be lunar ; to carrying a branch of myrtle, willow, pomme citron, and a bough of palm on Sukkot ; for allowing forty stripes minus one for the accused ; for prohibiting a kohen from marrying a harlot ; for prohibiting the testimony of women in court ; saying the Shema twice a day; placing a Mezuzah on your doorpost; and wearing tefillin .
Keep in mind that he wasn't from the Perushim sect by birth, in fact, he criticized them at first, and only after much study, came to the conclusion that what they taught was from Hashem.
III. THEY FOUND THE MISHNAH IN ETHIOPIA
For 2,400 years, the Abyssinian Christians have unknowingly kept a fringe of oral tradition. This is because they were once Jews who fled after the destruction of the Beit HaMikdash. In 1992, researcher, Graham Hancock, published his journal of discoveries in search for the lost Ark of the Covenant. One slice of his account recalls his meeting with an Ethiopian priest by the name of Memhir Fisseha. He recollected the way his people used to offer burnt offerings by collecting the blood of the lamb in a "bowl" - he used here the Hebrew "gomer," which means "finished" or "complete," and is utterly foreign to his native Ge'ez. The blood was then scattered over the stones of the tent, which, he claims, once housed the Ark. When asked to rehearse the action, Fisseha “positioned himself next to the stones with the bowl in his left hand, dipped into it with his right forefinger, swept his right hand above the level of his head and commenced an up-and-down motion. ‘The blood was scattered in this way,’ he said, ‘over the stones and over the tent of the Ark. Afterword, as I told you, what was left was poured thus.’ He then tipped the bowl sideways above the cup-shaped hollows in the tops of the pillars." It was as if, Hancock later reflected, he’d been wielding a whip instead.
Back in England, Hancock poured over ancient sources only to discover that Fisseha’s actions had fitted exact descriptions of the Kohen Gadol, as recorded in the Mishnah’s tractate, Yoma. Accordingly, the blood was collected and stirred so as to not be congealed. Then, the Kohen Gadol would arrive and take the bowl, sprinkling the blood “once upwards and seven times downwards [on the curtain outside, facing the Ark]. . . . as though he were wielding a whip."
Fisseha had never read the Mishnah, and yet, it seemed as if he knew exactly what it said .
IV. WHO ARE THE KARAITES?
They were basically a break-away movement (no relation to the צְדוּקִים) in the mid 760s CE. The Exilarch, Shlomo, was childless when he died, and so the Geonim, doubting his nephew's character (Anan), passed on his position to another. Anan rebelled, and was locked up by the Caliph, whom he later convinced to be set freed. Once free, he started his own interpretation of Torah, often taking on a hand's-down literal approach, such as not allowing a doctor to heal your illness (Shemot 15:26), which, unsurprisingly, resulted in his own death. It should be noted that the Karaite movement, while rejecting our Oral Torah, wear kippot and do other decrees only known orally.
Well, I think that about wraps. Proof of an Oral Law is easy to come by when you know what to look for, and of course, this short introduction isn't by far exhaustive. Nonetheless, I'm glad to have done my part. Reach out to me if there's more you'd want to know, the examples are literally never-ending.
See Bava Metzia 59a-b, B.T., for a talmudic reference and Devarim 20:14-20, 23:25-26, Vayikra 14:33-34 for these applications in action. Specifically see Rut 4:1-2 for a local beit din.
To name a few, compare Devarim 8:10 with Shmuel Alef 9:13; Vayikra 22:4 with Shmuel Alef 21:5, 28:3; Vayikra 13:46 with Melachim Alef 5:1-5, 7:3; and Vayikra 20:27 with Melachim Bais 4:23 and Amos 8:5.
Antiquities, Book 3, 10.2-5.
Compare Devarim 25:3 with Maakos 3:10 and J. Book 4, 8:21.
Compare Vayikra 21:7 with Kiddushin 4:1, Kesubos 2:5, J. Book 3, 12:2.
Compare Mishnah Shavuot 4:1 with J. Book 4, 8:15.
J. op. Cit. 8:13.
Hancock, Graham, The Sigh and the Seal: the Quest for the Lost Ark of the Covenant. A Touchstone Book published by Simon & Schuster, Inc. New York, NY (1992). 600 pp. pp. 141, 214-218.