Can Chinese words start with a vowel?
There are some Mandarin Chinese Pinyin sequences which consistently start with a vowel. As mentioned in the comments, these have a Pinyin representation which starts with y or w:


義, Pinyin yì, IPA /i⁵¹/ (starts off with /i/, the close front unrounded vowel)
完, Pinyin wán, IPA /u̯a̠n³⁵/ (starts off with a dipthong containing /u/, the close back rounded vowel)


The others which many people think start with a vowel but don't really except in certain (albeit extremely common) circumstances start off with a Pinyin letter a, e, or o. Pronounced in isolation or after the end of a pause, these do not start off with a vowel, but a glottal stop, which is a consonant that requires you to block airflow through the glottis.

For example, consciously take caution of the difference in how you pronounce 餓 in the following:


餓 (in isolation)
我餓了


In (1), 餓 must start with the glottal stop consonant (take notice of what you're doing in your chest area when pronouncing this), while not ordinarily in (2), although you can use a glottal stop there sometimes for emphasis.

Without this consonant, you cannot pronounce sequences like 諤諤 or 阿娥 properly (although these sequences aren't common at all). An English example would be the word uh-oh!, which must have the glottal stop at the beginning of the second syllable, else it would sound incomprehensible.