Believe it or not, email service providers actually exist.

Rollernet.us is a good one. They have excellent deliverability, reasonable prices, and everything you could want related to email.

They have a few minor other services, like DNS management, but they are not a cloud compute provider.

Another option is to use a cloud compute provider like AWS. You don't need to run the VM yourself to use SES for email messages. The hard part is the webmail access: you have to choose between a poor interface (an S3 bucket) or running a managed VM to host something like Roundcube.