Hey,
Your DEP connector should not have anything to do with the ability to deploy apps to your devices. That connector only links your Apple business manager or Apple school manager account with Intune so you can deploy profiles to your devices during the initial setup.
If you go in under Endpoint.microsoft.com -> Tenant Administration -> Connector status
what does it show you, are there any "unhealthy" connectors ?
If you are using VPP and buy apps through the Volume purchase program and that token is unhealthy that could cause problems with the App install. Also if the APN cert has expired that could also cause problems I believe.