Before a plain text email is sent in WooCommerce, all HTML tags and some HTML entities are stripped. Specifically, WooCommerce uses wp_strip_all_tags and preg_replace with a custom set of patterns and replacements to perform this operation.
You can inspect the code in get_content in includes/emails/class-wc-emails.php.
It appears that in this preg_replace operation, WooCommerce removes the HTML entity ¥ (¥) and thus, you are not seeing the currency in your plain text emails.
Note that $plain_replace includes the actual symbol for $ and £ and thus these currencies will not be removed.
Unfortunately, neither the pattern nor the replacement ($plain_search and $plain_replace) can be extended via a filter and thus your only option to exclude ¥ from being replaced would be to change the code inside the plugin. However, these changes will be overwritten whenever you update WooCommerce.
The most viable solution to exclude your currency symbol from being removed from the email content before the email is sent would be to overwrite your currency symbol in get_woocommerce_currency_symbols in includes/wc-core-functions.php using the filter woocommerce_currency_symbols.
In your case, you would replace ¥ with YEN.
However, this symbol is also used in all places where the currency symbol is displayed in the shop (e.g. cart, checkout, product page, etc.). That is, your website will display YEN instead of ¥ in all these elements.
This would be the most reasonable approach in my opinion for all currencies that don’t use a special symbol for their code (e.g. AMD, EGP, CHF, …).
In your specific case, where the “letter code” should only be used in the emails, I recommend extending / overwriting the WooCommerce email templates instead.
You should be able to do this without overwriting the actual email templates (and thus preserving upwards compatibility and after-update-issues down the line) by replacing the hooked actions with your custom actions (for instance woocommerce_email_order_details in customer-on-hold-order.php).