Can I use CloudKit on Android or web-based app?
Unfortunately CloudKit is only available for the Apple ecosystem. However, there are similar technologies called Parse and FireBase that allows you to do the same, but can be cross platform. However, they cost a lot more and are not client-side services.
NOTE - Parse's hosted services will be fully retired on January 28, 2017.
Yes you can. Apple provides CloudKit JS, specifically designed for web services. I don't know much about Android, but I'm pretty sure it'll be not a hard challenge to run JavaScript.
Also CloudKit WebServices could be interesting for you.
EDIT advice and discussion
To give you an honest advice: Better use something "own". I currently work with a custom server on an AWS EC2 instance and am really happy.
You could, for example, write a really simple server using Node.js and connect a Mongo DB NoSQL database. CloudKit is actually not more than this.
This is really a simple task. I did this before and with some JavaScript experience and a few days Node exercises it is absolutely feasible; you'll write really nice servers very quickly.
In the end, when dealing with more customers, CloudKit will be more expensive, actually. And if you, why ever, must move to a different service, you will have trouble with CK, because you are not able to access the privately stored data.
Also, be sure that CKs concept fits your needs. I was in your situation a few months ago. As I read more about CloudKit and viewed some WWDC sessions, I more and more realised that it is not a BAAS as you would probably expect.
One example: You have no access control: private or public, thats it. There is a public database which everyone can access each resource in. And a private one for any user, which is inaccessible by others.
If you don't want to, or can't, do something on your own, you could simply use BAASBOX for self-hosted APIs or just any commercial BAAS.
EDIT II
To point that out again for anyone late in the game:
The private (per-user) databases are absolutely inaccessible for others – even you as the developer and operator can't access the, most probably encrypted, data to move to a different (maybe self-hosted) service.
You'd need to make an update and the app then needs to move the data to your new service on the users behalf – "device-ly".
These kind of processes are typically problematic because you'd need to run two services until all active users moved their data – which is REALLY hard to tell; your customer might be idle for some time and they'd be upset if their data is lost in void forever.
Ok – Cloudkit might continue until... Yeah, until then. So it might not be as problematic to do that kind of passive longterm-movement.