I have a website hosted on a shared server at company X(BigRock/HistGator/GoDaddy etc.). They provide x number of email addresses as part of their hosting solution. I am planning to move my website to Azure and get rid of the hosting plan being used at company X. I would like to know how can I move my emails to Azure along with my website so that I can keep receiving and sending emails from my company account - email@example.com . If there's no way of moving the email server, what options do I have?
The only option is to build your own e-mail server using virtual machines which I don't recommend for your case. Microsoft e-mail solution is Exchange Online.
To send bulk marketing e-mails from your site, you can use SendGrid from Azure Marketplace.