MyWorks Sync does support syncing customers, orders, payments and more from WooCommerce to QuickBooks as part of our integration. We support syncing payments made in WooCommerce by almost any payment gateway that can be used within WooCommerce.
It's important to note that MyWorks is a sync/integration - not a payment processor/gateway. Although we support syncing payments made by any WooCommerce gateway you choose to use - MyWorks is not a payment processor.
Wondering what payment gateway/processor to use for your site? We recommend checking with your web developer first, as they are the best resource to both recommend and install/configure a payment gateway on your WooCommerce site.
The most popular payment methods we see used are Stripe, PayPal, and Authorize.net - but there are many other available gateways based on your preference.
Note: WooCommerce.com does offer an Intuit Payment gateway here - for users wishing to use their existing Intuit Payments account to process payments on their site. This is not affiliated with MyWorks, and any questions about setup/usage for this gateway should be directed to their plugin developers.