EZGif is best online solution IMO. They have a nice spritesheet cutter too.

I am sure there are plenty of downloadable softwares as well, I just prefer to use EZGif.

"However, I can't find anything I can use to animate the sprites into a gif. Any suggestions for programs like that?"

You can do that with the GIMP. I'm not saying it's easy, but perfectly doable:

1. Put every frame into a separate layer (first frame at the bottom, last on the top). There's an "Open File as Layer" in the "File" menu, or you can paste the content and press "Add New Layer" (this way it will paste it into a new layer).
2. To shrink the output gif's size, choose from the menu "Filters" > "Animation" > "Optimize (Difference)" (there's also an optimize for GIF option, but it doesn't really work in my experience. Optimizing for difference always works.)
3. From the menu, choose "File" > "Export As"
4. Give it a name which ends in ".gif" and press "Export"
5. In the popup window, check "GIF Options" > "As animation".
6. Press "Export" and voilá.

If you're not afraid of command line, then you can use ImageMagick, standard package on all distro. It provides a command "convert" which can convert a series of pngs into an animated gif like

convert -delay 100 *.png animated.gif

Another possibility is using ffmpeg. It is for videos by default, but it can be used to create animgif from a list of png files too.

ffmpeg -framerate 30 -pattern_type glob -i '*.png' -r 10 animated.gif

Cheers
bzt

oh. look at https://www.screentogif.com/ and https://getsharex.com/