What is the oldest inscription of Yahweh found?

This is only answerable with a certain amount of discussion. To illustrate the problem bluntly: you expect a date well before the exile, and "Yahweh" will be found absolutely nowhere there or then. There were no vowels recorded in the local scripts. And the biblical tetragrammaton YHWH (יהוה) is by far not the only form that most scholars believe to be one form of recording a/the name of a certain deity.

The short forms connected to the name of that deity are jw, jh, jhw, jhh and hjw.

The short form jw is perhaps found in the Ugaritic Baal epos (KTU 1.1 IV:13-20 Although the text would then emphasise that the son of Baal should not be called Jahwe but Jammu…). This would then date to ~1500 BCE.

The oldest reference outside the bible is still from the Egyptian inscription dated to the reign of Amenophis III. and a list from Ramesses II. were the land of the Shassu of Jahu (t3 š3św jhw3) are mentioned. This would be 1402–1363 BCE respectively 1279–1213.

This may very well be the same deity or a precursor form or a different deity with just a similar name.

The oldest unquestionable longform finds would be some shard fragments from Kuntillet Ajrud, although the significance of this find remains disputed.

The oldest unquestionable longform with significance is then probably the Mesha stele set up around 840 BCE.

As the longform name is then attested multiple times in non-biblical Hebrew epigraphy from the 9th century BCE onwards, it looks like the longform will not be found in the future with a date that is that much further in the past. Please be aware that predictions are difficult if they are about the future.

Current overview

shortform   
nonbiblical 
first time 
evidence as 
part of
names   BCE place       biblical examples

Jw-     950 Israel      [Jonatan][3]
-jw     950 Israel      Miknejaw
Jhw-    900 Judah       [Jehoschua][3]
Jhh-    900 Judah   
Jh-     700 Judah   
Jw-    ≈600 Egypt   
-hjw    500 Elephantine 
-jhw    300 Edfu        [Elijahu][3]
-jh     200 Transjordan [Sacharja][3], [Jesaja][3], [Hiskija][3]

In this answer the one factor apparently primarily motivating the question – Delitzsch's assertions – is underdeveloped. That is because this man displayed a huge amount of bias that motivated his findings. While he did make some interesting observations, he simply goes to far. In this case, he goes to far back to find supposedly corroborating evidence once his hypothesis was formed. Mostly based on opposition to his goals and consequences of this kind of thinking for theology and politics there are a few dispuation describing the Delitzsch Babel-Bibel controversy: WP: Babel-Bibel-Streit, WibiLex:_ Babel-Bibel-Streit and Reinhard G Lehmann: "Friedrich Delitzsch und der Babel-Bibel-Streit", Vandenhoeck & Ruprecht: Göttingen, 1994. (Online)

However:

Other Etymologies
But what if yhwh is not a part of the verb hyh/hwh (to be)? First, perhaps, we may return to the possibility—mentioned above in respect of interpretations of yhwh as part of the verb “to be”—of considering that the association is not one of etymology (perhaps rather a modern discipline). This would permit an understanding of the name not as an etymology, but rather as some form of paranomasia or wordplay. The association of the name with the verb “to be” might then remain at the level of the canonical text, with the origins of the name open for speculation. And such speculations are legion, given the number of languages which may be imagined as possibly original for the name: Sumerian, Akkadian¹, Eblaite, Egyptian (the most minimalist historians of Israel have found reliable data here), Phoenician, Midianite, Amorite, Edomite, North Arabian, and Indo-European languages.

_{Footnote 1: yhwh appears as a theophoric element at the beginning and end of Israelite and Judaean names mentioned in neo-Assyrian inscriptions. R. Zadok, Pre­Hellenistic Israelite Anthroponomy and Prosopography (OLA) 28 (Leuven, 1988); Weippert, “Jahwe,” pp. 246– 253; Weippert, Jahwe und die anderen Götter (FAT) 18 (Tübingen, 1997), pp. 35–44. Neo-Babylonian inscriptions have names with the theophoric element at beginning and end but spelled with a /m/, which conceals a /w/. So, J. Tropper, “Der Gottesname *Yahwa,” Vetus Testamentum 51 (2001), 81–106 (with bibliography). There is also onomastic evidence for the /h/ of yhwh: F. Joannès and A. Lemaire, “Trois tablettes cunéiformes à onomastique ouest–sémitique,” Transeuphratène 17 (1999), 16–33; L. Pearce, “New Evidence for Judaeans in Babylonia,” in Judah and the Judaeans in the Persian Period, eds. O. Lipschits and M. Oeming (Winona Lake, 2006), pp. 399–411. Earlier, an Akkadian etymology (ia­u, “noble one”) for yhwh was suggested by F. Delitzsch (a notorious advocate of Assyriological origins), Babel and Bibel (Leipzig, 1921), pp. 79–80. One might also consider Theodor Fritsch’s unambiguously titled Der falsche Gott Beweismaterial gegen Jahwe (Leipzig, 1910), which is at least anti-Semitic if not almost Marcionite. Cazelles, “Pour une Exégèse,” pp. 11–24, suggested on the basis of the Akkadian yau a meaning for the Tetragrammaton as “the one who is mine.” Previously, in idem, “Mari et l’Ancien Testament” in XVe Recontre assyriologique internationale, ed. J.R. Kupper (Liege, 1966), pp. 73–90, 82–86. The suggestion was previously made by Bauer, Landsberger, and Dhorme. Stephanie Dalley, “Yahweh in Hamath in the 8th Century B.C: Cuneiform Material and Historical Deduction,” Vetus Testamentum 61 (1990), 21–32, offers a clear explanation for non-specialists of the problems of recognizing yhwh in cuneiform material. Positively, she argues against the assumption that every bearer of a proper name compounded from yhwh was a either a Judaean or Israelite, as she considers Yhwh may have been worshipped in several Syrian cities. See also, A. Murtonen, The Appearance of the Name Yhwh Outside Israel (Helsinki, 1951).}

Robert J. Wilkinson: "Tetragrammaton: Western Christians and the Hebrew Name of God. From the Beginnings to the Seventeenth Century", Studies in the History of Christian Traditions, Vol 179, Brill: Leiden, Boston, 2014, p31–32.

Source: Karel van der Toorn & Bob Becking & Pieter W. van der Horst: "Dictionary of Deities and Demons in the Bible – DDD", Brill: Leiden, Boston, Köln, ^₂1999, P910–933.
Short German labguage overview by Becking: "Jahwe" (2006) on WibiLex

In any question regarding biblical archeology the current outlook profits from reading Bob Becking & Lester L. Grabbe: "Between Evidence and Ideology. Essays on the History of Ancient Israel read at the Joint Meeting of the Society for Old Testament Study and the Oud Testamentisch Werkgezelschap Lincoln, July 2009", Oudtestamentische Studiën, Vol 59, Brill: Leiden, Boston, 2011.