Is standalone APM server required with Elastic APM Integration?

Elastic Agent would set up a local APM Server if it is enrolled with to a policy that contains the APM integration, so localhost:8200 should be valid on such hosts.

If you go to Management -> Fleet, you can observe agents, policies and other settings. In the Agents tab, you should see a list of your agents. Do you see your agent? Does it have a "Healthy" status or other? You can get further info on what's going on if you click on the agent and switch to the Logs tab, where you can see all agent-related logs. By default, only the root agent logs are shown, so you should mark the apm_server dataset in the Dataset dropdown.
Otherwise, you can login into the host and collect logs from the agent installation folder.
Hopefully this is at least somewhat helpful