Yes, you can use third-party email hosting services for your Shopify store, you have to because Shopify does not offer hosted email.
You have several options on how you can set it up depending on who your domain name provider is, it’s just a matter of making sure your domain has the correct cname code. This page will give you a little insight on how to go about setting things up for email. https://help.shopify.com/en/manual/domains/email-forwarding-and-hosting/email-hosting
First and foremost you will need to make sure that your domain name is verified with your email hosting provider by adding the proper TXT code to your Domain Name DNS file.
These directions will help if you utilize Google for email however using the same basic steps will allow you to setup email with just about any provider as long as you have access to your domain name DNS file file. I utilize a GoDaddy subsidiary for my email and it works fine. Hope this helps.
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Since Shopify does not offer email you must use a third-party service for your business email. You can use any email service as long as you own the domain name for the email that you are attempting to set up. You must have access to the Domain Name DNS file because this is where you set up your email TXT file.