I too faced this issue.
Problem: Option "AWS: Deploy SAM application" is missing in VS Code command palette despite of AWS Toolkit installation.
Solution:
In VS Code AWS toolkit explorer, I see directory structure (e.g., API Gateway, IOT, Lambda, S3 etc), right click and and pick 'Sync SAM application', ignore message 'The SAM CLI will use the AWS Lambda....', just click on Ok and follow instructions in VS Code command palette.
Note- You should have a S3 bucket created yourself before performing above action as you will be prompted to select a s3 bucket in command instructions.
Also wait for 1 minute to get the sync process completed.
It worked for me.