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I wasn't able to find anything about this in 5th Edition. A drake seems that it would exceed the 1/4 challenge rating stated in the PHB.
In this specific campaign, our Dungeon Master allows anything that is accessible on D&D Beyond not including Homebrew content.
I did find the Drake Warden Ranger archetype, which I assume is for Pathfinder, and thought it might be possible that I’d overlooked a similar variant in D&D 5E.
Drakes, though animal like in appearance and behavior, have the creature type of dragon, not beast. And the Ranger's Companion states:
Choose a beast that is no larger than Medium and that has a challenge rating of 1/4 or lower (PHB, p. 93 bold added).
"Beast" is not just a descriptive word here, it is a creature type (see Monster Manual page 6) that applies to some creatures and not others. If a creature is a beast, it will say so in its stat block (right under its name, right after its size).
There are some interesting examples of intimidating (read: awesome) creatures that are 1/4 CR beasts: a pteranodon springs to mind. But drakes are not beasts, so they would not be eligible as a Beast Master Ranger's Companion (even if a CR 1/4 one existed in some published adventure).
As of October 2021 (almost exactly three years after this question was asked), a new Ranger subclass has been published (in the book Fizban's Treasury of Dragons) called the Drakewarden. This class grants the Ranger the ability to summon a Drake companion at 3rd level. The statistics for said drake can be found in source book mentioned above. This subclass is worth looking into if you are invested in a drake as your companion, since (as mentioned before) the Beastmaster Ranger cannot have drakes as their companions.
The lowest-CR drake on D&DBeyond has CR = ½
A ranger's companion must have CR ≤ ¼.
So no, you cannot (currently) have a drake as your animal companion. To do so would require either a modification to the ranger rules or a new drake with CR ≤ ¼ (and the beast type, as gandalfmeansme rightly points out) to be published by WotC.
