Connecting a hose to a 3/8 compression fitting
Not knowing what your valves under your sink look like , I would suggest possibly a new valve with 2 ports or the addition of a T  on a 3/8 fitting then you could find a pre made hose like you want for a Refrigerator, this can feed your waterer. 

I would want an independent shut off incase of problems, 
my plumbing speciality store could set up all the parts needed and suggest things like an extra valve just in case by using a photo of your existing under sink set up. They may cost a few dollars more but 1 trip ends up saving time and frustration in the long run compared to the big box stores , the big box may have what you need but getting the correct help sometimes it tough.