Can I give players capes?

Solution 1:

You want Cape Mod. The player will need to upload their cape for you to see it, and only people with the mod will be able to see capes.

Solution 2:

Not legally

Disregarding any and all technically possibilities of adding capes to players, it's disallowed by Mojang. They explicitly asked people not to give capes to players in a Mojang Blog post about server monetisation from June 2014:

You are allowed to sell in-game items so long as they don’t affect gameplay
[...]
There is one exception to this rule – capes! We have a lot of fun making cool capes for extra-special members of our community and Minecon attendees. We’d like to keep them as exclusive as possible. So, yeah, no capes please, even if you’re giving them away for free.

(emphasis mine). Capes are meant to be highly special, given out to only the most loyal fans and members of the community.

Futher reading: Minecraft Commercial Use