Where can I find the earliest surviving versions of Herodotus's Histories?

There are early 1500's editions of Herodotus: see the wikipedia article about their publisher Aldus Manutius. That article refers to

Works published from the Greeks. Manutius printed thirty editiones principes of Greek texts, allowing these texts to escape the fragility of the manuscript tradition.

and the wikipedia page about Herodotus has an illustration of the kind of fragility in question. (But not of a bit of writing Aldus used.)

Going further back in time: my local uni's library has a copy of a 1475 Latin translation, printed by Arnold Pannartz. The 1911 Britannica article on H says

The history of Herodotus has been translated by many persons and into many languages. About 1450, at the time of the revival of learning, a Latin version was made and published by Laurentius Valla. This was revised in 1537 by Heusbach, and accompanies the Greek text of Herodotus in many editions. The first complete translation into a modern language was the English one of Littlebury, published in 1737...

But of course as many commentators here have pointed out, these printed editions appeared about 2,000 years after Herodotus lived. Nothing like a "primary source" (whatever you mean by that) survives.

For a critical discussion on how modern versions of Herodotus' Histories have been compiled from the surviving manuscript sources, you could do a lot worse worse than the 1983 essay On Editing Herodotus by R. A. McNeal.

This includes a useful overview of the merits of the various surviving copies and partial copies of the text.

Essentially, the problem is that no totally complete manuscript of Herodotus' Histories survives. Furthermore, none of the early manuscripts that do survive completely agree on the text (mainly due to scribal copying errors and selective editing in medieval scriptoria). The earliest surviving copy, "called 'Codex A", is owned by the Biblioteca Medicea Laurenziana in Florence and forms the basis for all modern translations.

The principal surviving manuscripts for Herodotus' Histories are assigned letters. The earliest examples (dating to the 10th - 14th centuries) are:

• A Florence, Laurentian 70, 3. - 10th Century CE
• B 'Codex Angelicanus', (named for the library in Rome where it is currently held, but previously known as the Passioneus manuscript). - 11th Century CE
• D Vatican graecus 2369. 11th / 12th Century CE
• R Vatican graecus 123. A composite manuscriptdating to the 14th Century CE
• S The "Sancroftianus" (named for its former owner, Archbishop Sancroft). 14th Century CE
• U Rome, Vatican Urbinas 88. 14th Century CE
• C Florence, Laurentian conventi soppressi 207. 11th Century CE
• E Paris, BNF suppl. 134. (Also contains extracts of Plutarch and Diogenes Laertius). 13th Century CE
• P Paris, BNF gr. 1633. 14th Century CE

In addition to the manuscripts, we have a number of much earlier fragments of the texts written on Papyrus. None of the papyrus fragments contain the full text. Most contain just fragments of single pages. Most are part of the collection of fragments known as the Oxyrhynchus Papyri, for which a searchable database is now available online.

I haven't been able to find images for most of these online, but the main examples that I'm aware of are:

• P.Oxy 1619 - late 1st Century CE
• P.Oxy 1375 - 1st-2nd Century CE

(P.Oxy 1375 - 1st-2nd Century CE - Image source - Photographic Archive of Papyri in the Cairo Museum)

• P.Oxy 2099 - Early 2nd Century CE

(P.Oxy 2099 - Early 2nd Century CE - Image source - Wikimedia)

• P.Oxy 1092 - 2nd Century CE
• P.Oxy 1244 - 2nd Century CE
• P.Oxy 2095 - 2nd Century CE
• P.Oxy 2097 - 2nd Century CE

(P.Oxy 2097 - 2nd Century CE - image source - Oxyrhynchus Online)

• P.Oxy 2096 - Late 2nd Century CE
• P.Oxy 2098 - Late 2nd Century CE
• P.Oxy 19 - 2-3rd Century CE
• P.Oxy 18 - 3rd Century CE
• P.Oxy 695 - 3rd Century CE

Other important papyri include:

• Munich papyrus A (See Archiv für Papyrusforschung vol. 1, p471) - 1st-2nd Century CE
• British Library Papyrus 1109 1st-2nd Century CE
• A Rylands Papyrus (Catalogue of the Greek papyri in the John Rylands library, vol. 1, p180) - 2nd Century CE
• Amherst papyri II, p3 - 3rd Century CE
• British Library (Milne No. 103) - 4th Century CE

The images above illustrate just how little of the material is actually found on many of these fragments!

EDIT

Thanks to @kimchilover for finding the image of P.Oxy 2099.