I actually track all this information myself for my various holdings. However, I do not use vanguard, nor any of the others listed ,but have to wonder if there is some type of transaction log. If you know two points, lets say Price per share and total invested at the time of purchase, (or some assortment of that) you should be able to determine shares purchased. Or at least get very very close to that number. Rounding is a thing and is common in the industry or so I have found in my records.
Otherwise yea, your going to have to go off the quarterly report, which isn't terrible but doable.
In all honesty I am surprised you don't have access to this information somewhere on your account page. It strikes me as odd that you can't see total shares owned at any time. If you could, then you should again with little effort be able to do the math and get what you want after every purchase. No it's not as nice as a .csv file, but most people don't care, as has been stated. They like to see the $$$ balance.