To my understanding, vātēs "bard" started out as an i-stem noun, built on the stem vāti- (probably from something like *weh₂t-i-). So I would expect the nominative to look something like *vāti-s.
However, we don't see *vātis—we see vātēs.
Where does this come from? I'm not familiar with any Latin sound changes that would turn short i into long ē in an unstressed syllable.
(Inspired by Sumelic's comment on another answer.)
It's not just vatēs - see Weiss pp 243-244 for details. He mentions 30 i-stem hysterokinetic nouns that have -ēs ending in nom.sg. (instead of the expected -is), for example: aedēs, caedēs, cautēs , cladēs etc.
I am aware of two proposals.
Analogical leveling in Latin
Some researchers have claimed that nom.sg. -ēs is due to analogical leveling after acc.sg.
e.g. Meiser 1998/2010 "Die Nom.-Endung -ēs bei Substantiven wie aedēs sēdēs vulpēs vatēs u.a. ist statt ererbtem *-ē(i̯) => *-ēi̯s > *-ei̯s > †-īs analogisch zur Akk.-Endung -ēm gebildet worden" (p. 142)
or Weiss "it may be analogical after acc.sg. *ei̯-m̥ > *ei̯em > *-ēm > -em" (p. 243).
Accent-ablaut classes in PIE
Weiss says it could also be a regular outcome of PIE [? - Alex B.] *ēi+s (p. 243),
cf. Vine 2017: "some 3rd declension nouns in nom.sg. -ēs may derive from PIE hysterokinetic forms in nom. sg. -ḗi̯ or -ḗn (> pre-Lat. *-ē) (p. 758)
or Klingenschmitt "Der Nominativ Singular, welcher im Urindogermanischen auf -é <-ḗi̯ ausging (vgl. av. a. kauuā, j. kauua < *kou̯h1é), wurde in einer Vorperiode des Italischen durch Anfügung der für den Nominativ Singular eines Großteils der nicht-neutrischen Stämme charakteristischen Endung *-s zu *-ēs verdeutlich. Dieses Paradigma hat in einem Vorstadium des Lateinischen offensichtlich auch Wörter aufgenommen, die - von Haus aus zu einer anderen Stammklasse gehörig - im Nominativ Singular auf *-ēs auslauteten" (p. 114).
If you read earlier literature (i.e. before the theory of the accent-ablaut classes gained its popularity/acceptance), e.g. Leumann 1977 (pp. 343-344) or Tronskii 1960 (p. 157), they used to say that the grouping of such nouns as well as its origin was unclear,
cf. Ernout 1952 (2014) "l'origine de cette finale n'est pas exactement connue, mais elle semble ancienne" (p. 47)
or even Sihler 1995 writes "The history of these nouns is unknown" (p. 316), cf. §322b on page 334.
The OLD mentions two occurrences of nom.sg. vatis: bonus vatis (Plautus, Mil. 911), although the 2011 Loeb edition (by Wolfgang de Melo) has uates, and Cicero Div. 2.12, cf. the 1923 Loeb conjecture (by W. A. Falconer) "Num igitur aut, quae tempestas impendeat, vates melius coniciet quam gubernator?"
