The fixed phrase "vor allem" is an adverb and can be replaced by e.g. "besonders", "insbesondere", "in erster Linie" or "vor allen Dingen".
See e.g. https://www.wortbedeutung.info/vor_allem/ or https://de.wiktionary.org/wiki/vor_allem
For this meaning you have to use the fixed phrase unchanged, independent from the case or number of the noun following it.
The example sentence
Vor allem Frauen sollen dafür qualifiziert werden.
could be translated to English as
Especially women should get qualified for this.
emphasizing that it is particularly important for women.
The adverb "vor allem" can not only emphasize a noun but also other words like a verb, adverb, adjective.
Modified examples:
Frauen sollen vor allem dafür qualifiziert werden. (emphasizing "dafür")
Frauen sollen dafür vor allem qualifiziert werden. (emphasizing "qualifiziert")
The other example
Vor allem Jugendliche sind immer auf der Suche ...
is correct. Your wrong translation "vor alle Jugendliche" would not make sense in this sentence without other changes because this would turn the subject "Jugendliche" into a part of an adverbial "vor alle Jugendliche".
If instead of "allem" you use a form that matches the noun like
Vor allen Frauen ... (Dativ, location, static))
or
Vor alle Frauen ... (Akkusativ, direction, change)
it is no longer a fixed phrase but a combination of a two-case preposition (Wechselpräposition) "vor" and an indefinite pronoun (Indefinitpronomen) "alle". Depending on the context, this would change the meaning to
In front of all women / before all women ...
This would not make sense in the example sentence without other modifications as already explained above for your other example sentence.