Yes. WooCommerce runs on WordPress, so a store sits naturally on Lynx managed WordPress hosting (WP Engine) or Cloudways managed cloud, and hosting can be bundled with a build.
Yes. WooCommerce is WordPress-native, so a store runs on the same hosting we use for any serious WordPress site: Managed WordPress Hosting on WP Engine (from $25 a month) for most stores, or Cloudways Managed Hosting (from $49 a month) when you need scalable, fully managed cloud infrastructure. Optional WordPress Monitoring ($4.95 a month) keeps the store patched, backed up and checked for malware.
If we build the store, hosting and domain can be included in the project. If you already have a WooCommerce store elsewhere, we can host it too. See eCommerce Websites and Web Hosting.
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