Actually, neither א nor ע are vowels, and they both modify the sounds of letters after them. They are two different types of "glottal stop." You won't hear them so much here because Modern Hebrew is a compromise between the Ashkenazi (German/Eastern European) accent and the Sefaradi/Middle Eastern/Mizrahi accent. But in my Jerusalem neighborhood I hear both because I have friends on both sides.
The ayin ע is the stronger of the two, more gutteral, coming from the back of the throat. The alef א comes from the front of the mouth. But they both are glottal stops.
The British English use of glottal stops as a substitute for the "t" sound is explained here: https://www.youtube.com/watch?v=sHaN4jHBGVk
I'm looking for an explanation of how to apply this in Hebrew without patronizing or trivializing the speakers who make use of it. In the meantime, in my experience, native Hebrew speakers usually pause slightly before pronouncing a syllable that has an alef or ayin in it. I hope you will get the privilege of hearing a Sephardi/Mizrahi (it's not just for Temanim/Yemenites anymore!) pronounce these letters!