Run SQS on AWS SAM locally. Is it possible?

Solution 1:

Only in a limited sense and in no way "native". You can use SAM to locally test Lambdas that make outbound SDK calls to cloud-deployed SQS queues. You can invoke local Lambdas with mock triggering events to simulate invocation by queue messages.

See this answer to a related question, skipping point #1, which is specific to CDK applications.