Ok, I found a work-around that works just as good, it functions and it’s still easy for the end-client to update.
The problem seems to lie within Advanced Custom Fields’s oEmbed function. First, I created a field with the type of “oEmbed” (PRO-version only), where I would later in the post just copy and paste the vimeo-link of my choosing. And as I stated before, it fetched the video around 50% of every on-site refresh.
However, as I looked around and realized that WordPress do have these embed features already built into it; I thought I could just as well create another field, but with the type of “Wysiwyg Editor” instead, and paste the url in that field and see what happens.
The result: I’ve loaded the page around 50 times now and it loads correctly every time.