van der Waerden's original proof
You can look van der Waerden up at:
https://zbmath.org/?q=py:1926+se:00003568
but you only get 3 articles for free.
There's an article covering vdW's theorem at:
https://homepages.warwick.ac.uk/~masdbl/vdw.pdf
His paper is covered by Khinchin 'Three Pearls of Number Theory', which might contain an original translation:
http://store.doverpublications.com/0486400263.html
Digitized versions of the first and second series of the journal "Nieuw Archief voor Wiskunde" are available at Delpher, an open access site hosted by the Royal Library of the Netherlands (Koninklijke Bibliotheek).
The specific paper of B. L. van der Waerden[1] that you're looking for that appears in the second series of this journal can be accessed at the PURL: https://resolver.kb.nl/resolve?urn=MMKWG01:022157001:00228. While this isn't a translation of the paper, having the original paper in hand is definitely an asset. Google Translate is decent enough that you will probably be able to understand most of the paper with its help.
Instead of a translation of the paper, you could also do as Andrés E. Caicedo suggests in the comments and take a look at the expository article written by van der Waerden himself in 1954 about how he arrived at the proof of Baudet's conjecture[2]. This article was translated from German into English and published in 1971[3], and was republished in 2009[4]. One of these versions might turn out to be more easily accessible. You will also find several of the reprints of this article indexed at zbMATH Open.
- van der Waerden, B. L., Beweis einer Baudet'schen Vermutung. (German) Nieuw Arch. Wiskde. (2) 15, 212–216 (1928). JFM 53.0073.12
- van der Waerden, B. L., Einfall und Überlegung in der Mathematik, 3. Mitteilung: Der Beweis der Vermutung von Baudet. (German) Elem. Math. 9, 49–56 (1954). MR0061079, ZBL 55.03802.
- van der Waerden, B. L., How the proof of Baudet’s conjecture was found. Studies in pure mathematics: Papers in combinatorial theory, analysis, geometry, algebra, and the theory of numbers presented to Richard Rado on the occasion of his sixty-fifth birthday (Mirsky, L. (ed.)), Academic Press, London, 251–260 (1971). MR0270881, ZBL 241.05009.
- Soifer, A., Ramsey theory before Ramsey: Van der Waerden tells the story of creation. In: The mathematical coloring book: Mathematics of coloring and the colorful life of its creators, New York, NY: Springer (ISBN 978-0-387-74640-1/hbk). pp. 309–319 (2009). MR2458293, ZBL 1221.05001.