My customers aren't receiveing payment method update emails

As these emails are sent directly by Shopify, then the issue is probably happening somewhere in Shopify or in the email client of the customer. It could be that the customer's email provider is rejecting these emails. 
Having said that, can you try to send this payment method update email directly from Shopify admin > Customers > click on your customer > click edit button next to the payment method > click three dots next to the payment method > click "Send link to update card" button. 
If the customer still doesn't receive it, then it is recommended to double check your email domain configuration in Shopify and ensure that the emails sent from Shopify are being delivered to your customers.