when I click the 'Download' button for "animated" stuff it opens as a .webm file

There are some places you can't post those but you CAN post animated GIFs.

Is there some way to download these directly in GIF format, or would I need to run this through some webm>GIF software every single time?

All videos are, by default, uploaded as .webm format to the site.
If you want it in any other format, you will need to download it first then manually convert it yourself.

Alternatively, you can take posts from type:gif animated, where animations are uploaded in .gif format.

The animated tag on its own just means the post has an animation. There's 4(?) ways a post can be animated: a webm video file, an animated png (apng) file, a gif file, or a flash (swf) file. When you click "Download", it gives you the file that was uploaded to the site. You need to download and convert it if you want it in a different format (and be aware, for longer and larger animations, gif files will be huge and have a lot of dithering artifacts).

GIF is lossless image format that is limited to 256 colors, meanwhile WebM is lossy video format.
In addition to this, e621 will host files exactly how they were uploaded here, meaning pressing download will give you the file how it was as well, only difference being filename changed to MD5 hash. I have no idea why someone would be thinking that download link on webm post would give gif?

Let's say that you want two posts from the site as GIF which are WebM.
First one is hand drawn black and white, 500px and second long small animation. This will most likely convert into nice GIF regardless of what you do.

However the second one is full color and high framerate 3D animation, 1920px and 10 seconds with sound. Now we have a lot of issues... Because there's much more colors, we have to generate palette to get most used colors and then utilize correct amount of dithering to get rid of color banding without adding too much noise, additionally because the resolution is high this introduces more artifacts and bloats filesize a lot, so we might need to downscale. GIF operates with frame times rather than framerate, so maximum frame time is technically 20ms, which turns into 50 FPS, so we most likely need to half the framerate down to 30 FPS. With all that done, we are now still left with 100MB GIF file because of the lenght of the animation, so we need to go back and cut either resolution or framerate more or cut the lenght of the animation.

Because of this, e621 doing automatic GIF conversion would be insanity and not that useful. Services like Gfycat and Redgifs do convert animations uploaded there into WebM, MP4 and GIF, however if you go there and download the GIF conversions, they are always extremely tiny and laggy, e.g. with post #2471272, this is how better quality GIF conversion looks in redgifs. Some other sites like gelbooru will do filetype conversions to ugoiras, because zip full of JPG files is non-standard formatting, but even they convert to webm, not GIF.

Because of the differences between animation formats (unlike pictures where you should simply use PNG), we do allow seperate filetypes and filetype conversions to be uploaded, but they do have to make common sense (e.g. converting GIF to Flash is almost always nonsense) and done with at least basic level of knowledge instead of just using online converter. Some artists and creators will also post GIF version to FA or in description, so it's worth to give source a look.

Also you could, I dunno, just link to the post URL?
Never fully understood people on discord who download all everything from twitter, then spam it and can't tell where it was originally from.