Does Exchange Server (in-house) suppport REST API?

Regarding to the Exchange Team Blog, the REST API is also supported in On-Premise-Scenarios beginning from Exchange 2016 CU3.

Please see https://blogs.technet.microsoft.com/exchange/2016/09/26/on-premises-architectural-requirements-for-the-rest-api/ for reference. I've tested it with our Exchange 2016 CU5 (Non-Hybrid)-Deployment and it worked well.

You can for example query the mails from your account via the following PowerShell-Code:

$restdata = Invoke-RestMethod -Uri "https://YOUR-Exchange-Server/api/v2.0/me/messages" -Credential (Get-Credential)
$restdata.value 

The only REST API exposed by Exchange on-prem is for accessing user photos. All of the other REST APIs are meant for Exchange Online only, meaning in the context of an Office 365 tenant.