The sad answer is no.
It gets into how beer is made. Grains are soaked in warm water so enzymes in the husks convert grain starches into sugars. The water is drained, and the resulting sweet wort boiled with hops and possibly other ingredients to continue to build up the desired flavor profile. Another reason for boiling is the “hot break”, which is a point when proteins and other materials in the wort precipitate. It’s very obvious when this happens, and all that crud floating around in your beer can easily lead to a hazy finished product.
And to make matters worse, as the wort cools, there is a second chance for proteins and other stuff to make a beer hazy. Much of the “cold break” ends up at the bottom of the primary fermenter in a nasty layer of sediment call trub, but there is still going to be particles too small to sink to the bottom floating around making the beer hazy.
That’s why most brewers and breweries add something called finings to both the boil and near the end of fermentation. There are a number of different finings, but their general properties are they contain large, positively charged molecules that attract the crud that’s now floating in your boiling beer. Like a magnet in a sandbox, the finings providing a nice, heavy place for the break to accumulate for easier removal. And unfortunately, several common fining agents are derived from animal sources. Gelatin, derived from animal bones, and isinglass, derived from fish bladders, are both commonly used to clarify beer.
As a quick aside, the “beechwood aging” that Budweiser used to advertise had nothing to do with flavor. Pieces of beechwood bark were added to the fermenting beer to help clarify it.
How do you know if your favorite beer is vegan or not? Luckily the Internet is full of websites that list vegan beers, including the nicely complete Barnivore. Lucky for me my favorite brands, Sierra Nevada, Stone, and Anchor, are all vegan. Unfortunately Guinness is not.