"Il y a de petites vestes."
Translation:There are small jackets.
Perharps this link may clarify (or not) this matter. I transcribe below just a paragraph of French Vocabulary: Clothing. https://www.thoughtco.com/french-clothes-vocabulary-les-vetements-1371475
"Un Costume: Men's suit. (...) Une veste: a blazer, a sports coat. Un gilet : a vest - watch out for the false cognate - “un gilet” is the 3rd part of a 3 piece man suit, the one you wear under the jacket ≠ “une veste” which is the blazer part. “Un gilet” is also a cardigan sweater. “Un gilet” is also called “un veston” but it’s old fashion."
Further transcription and link: "The word vest comes from the French word ‘veste’ which means jacket, sport coat, the Latin word vestis and the Italian word vesta which means a robe or a gown. In very simple and general terms, a vest is a garment that covers the upper part of the body and is sleeveless. It is also used to refer to a sleeveless under-jacket in some places."
I would suggest the following translation: "There are small vests."
Your own citations above say:
Une veste: a blazer, a sports coat
the French word ‘veste’ which means
jacket, sport coat
ie neither source indicates that "veste" equates to the English "vest".
The first source also says, as you cite it:
Un gilet : a vest...the one you wear under the jacket ≠ “une veste” which is the blazer part.
which specifically indicates, as I read it, that vest... ≠ (does not equal) "une veste".
Note: in American English "vest" = "waistcoat" whereas, in the UK it is an undergarment. The French Vocabulary article seems to be using the American meaning (= "gilet").