Dear All,
We are creating a platform like Wix where the user will registered a domain with us and further there website will be ready with few steps.
My questions is do let’s Encrypt provide any APIs where we can install the ssl with it.
Any there any limitations?
Do we need to make any cost ?
Welcome to the Let's Encrypt Community, Syed 
There are various implementations of control panels that allow acquisition and installation of Let's Encrypt certificates. Any number of existing ACME clients can also be integrated into your deployment.
It's also a good idea for hosting providers to look at
which provides some helpful tips and answers some common questions. One important tip there is to subscribe to the API Announcements category on this forum, where you'll see infrequent posts about technical and policy changes to Let's Encrypt services that may affect users (for example, an announcement when a client software upgrade is going to be required).
Let's Encrypt's certificate issuance interface, using the ACME protocol (https://en.wikipedia.org/wiki/Automated_Certificate_Management_Environment), basically is a public JSON API that you can use in your own applications. Some of the projects linked in @griffin's answer are libraries and applications that interact with it in various ways. You might have to do your own software development work to integrate it with your environment if no existing client does exactly what you want. (People on this forum can help answer concrete questions about your integration, but probably not plan it in its entirety or develop the software for it.)
There's never any charge for Let's Encrypt certificates, but there are issuance rate limits
If these limits are too restrictive for your service, you can request an adjustment to the limits applying to your account or domain by following the procedure mentioned there.
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