I don't think there are any existing pets that will be removed come Legion. There might be a few Draenor ones that will be more annoying/tedious to collect, but nothing completely removed.
One pet to definitely add to your collection is the Felbat Pup
(not yet available). We'll be able to purchase it once the Demon Invasion World Event begins (early August). Developers have noted that it's an event-exclusive pet
. Demon Invasions end once the expansion goes live so don't miss out.
Fortunately, it's cageable - we'll probably see it on the AH post-Legion launch, but I would still take part in the event just in case!