Why does French use du for the masculine partitive instead of something like de le, while the feminine partitive looks like de la?
Also, does de l’ apply to both masculine and feminine nouns that start with a vowel?
Yes, de l' applies regardless of gender.
For du, French commonly shows an /l ~ u/ transformation, particularly when /l/ is "darkened" at the end of a syllable and particularly before consonant clusters, as in loyal ~ loyaux and such pairs. Basically, the approximant constriction of the /l/ is not far from that of /w/, which leads to a diphthong with the previous vowel that eventually monophthongizes.
de le does not have such an environment, but it does have two schwa sounds just waiting to be reduced. In Old French they were reduced to del. This syllable before words beginning with consonants was the environment needed for /l/ to become /w/. Wiktionary claims that this not ending up as deu / deau but as du is because of its being unstressed and thus "doubly reduced", losing the first component of the diphthong and being left with only /w ~ u/ for the monophthong (which later shifted to /y/ as it is today).
de la did not undergo this because unlike /e/ which became /ə/ over time, /a/ remained more stable. It's elided in fewer circumstances.
A parallel thing happened with à le to produce au.