Is it possible to join a local windows machine (on premises) to a Windows Server VM that is hosted on Azure Cloud?

I am setting up a Windows Server VM on Azure cloud and configure Active Directory Domain Services and Group Policy Objects. I am trying to join a local computer (on premise) to that Server and to apply the group policy that I configured on Windows Server VM. Would this be possible, or do I really need a on premises active directory server to join a local computer ?

It's possible, you can deploy ADDS server role, join domains and many other things on Azure as what you can do on-premise. In this case, you have to do three steps:

1. Deploy a P2S or S2S VPN connection to make sure the connectivity between on-premise and Azure side. Refer to VPN Gateway.
2. Deploy a DNS server and DC on Azure VM. Set the custom DNS servers on the Azure VNet as your custom DNS server private IP address and also make sure the DNS query on the local computer could resolve the DC Server. You could deploy the DNS server in Azure VNet before setting up a VPN connection so that the virtual network connection setting gets the update on all connections.
3. Once the VPN is set up, you could join the domain.

Alternatively, you could use Azure AD to create an Active Directory domain in the cloud and connect it to your on-premises Active Directory domain. Azure AD Connect integrates your on-premises directories with Azure AD. In this scenario, you could have an on-premises Active Directory domain first.

Moreover, you could refer to this article to choose a solution for integrating on-premises Active Directory with Azure.