We are sorry that you are having issues viewing the Vimeo videos on your Kindle. I do not own a Kindle so I have not tested the site on one, but I have been doing some research for you and pray that I have found some solutions to your current issues. I found these solutions on Vimeo’s help forum.
Here is the first:
Tap the Menu button > Settings > Clear cache.
If clearing cache didn’t help, go into Settings again and uncheck Accelerate page loading.