We have a "traditional" dns service, where some outsourcing company manages our dns services. We host a lot of services in the cloud (mostly Azure) and then must submit tickets to the dns guys to create cname -registrations (and sometimes also txt records for domain ownership validation) pointing to the services.
E.g. a web service in Azure could be at the address ourapplication.azurewebsites.net and today we would create a cname such as ourapplication.ourcompany.com pointing to that.
A lot of these services could very well be hosted in a subdomain such as ourapplication.app.ourcompany.com.
In Azure there is service called Dns Zone, which to my understanding is a full dns service. Can we keep our current dns setup for the main ourcompany.com domain but have dns services for *.app.ourcompany.com in Azure?
You always can delegate your sub-domain to a different set of nameservers using NS DNS resource records.
See this question for details.