This file displays with a "transparent background" but it's connected into the file, and I am unable to separate it out in illustrator due to how it is set up.
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Transparency with such illustrator files works only against colour backgrounds.
You see my red rectangle works fine with the bubbles. To the left and the right of the rectangle I have an empty background (shown in white, but there is no background fill so what you see is the default visual of a 100% transparent artboard. I have switched off the visibility of the chequerboard and a grey background from the original visual. Here is the layers' palette:
Unfortunately, I required a fully transparent stock bubble, so having a colour background doesn't help.
I do not know what you are intending with your file, but all transperent effects that I've seen so far work only against a background. That background may be someting complex like a drawing or a photo, but the effect cannot exist by itself.
I was looking for a transparent PNG to use as an animated effect on a website
I'm pretty sure AdobeStock doesen't sell those types of files. I think you're going to have to take what you already have and make it into what you need. It is possible. Just have to take the time and play around with the functions. After you make a workable file in Illustrator, you then bring it into PhotoShop and fix it in there. Then you can save it as a PNG. 🙂