Ash here from the Shopify team.
It's great to hear you're choosing Shopify for your ecommerce solution! Managing a way to accept payments from customers is definitely an important step when setting up an online store. As per our Terms of Service for Shopify Payments, this payment gateway wouldn't be an option in your case, due to the nature of the products you sell. While I can't recommend a specific third-party gateway that supports your products, I am happy to share this list here, which contains all of the major supported payment gateways for Shopify per country. On that page, you can select your country of operation to see the options available to you. I suggest checking out some of the available options, and then contacting their teams to check if they can support your products.
As for differentiating who can purchase the different products on your site, there are a few recommendations I can offer.
One possible solution is to use an app like EasyLockdown or Locksmith. These applications are capable of locking down certain sections of your site, allowing you to choose exactly which customers have access. Using this method, you could create a collection of your prescription products, or even lock your individual prescription product pages, and apply lock rules to them. You would then be able to give just certain customers access to those locked pages. Once they have access to the locked pages, they can purchase products from there as they normally would on your site.
Another type of application I suggest checking out, is one that gives you the ability to limit how many products a customer can order at a time. This will be handy to make sure prescription eligible customers can't over-purchase their prescription products. Here are some top choices to consider:
If you're not interested in using applications, another solution would be creating draft orders for your customers who are eligible for prescriptions. If you go with this method, you can create draft orders for your prescription clients, and send invoices directly to them for payment. Once they receive the invoice, they can complete payment for the order, and it will become a regular order in your Orders tab in the admin. You can then fulfill the order as you normally would. With this method, you can fully control who received a prescription order, and how many prescription products are in each order.
Another consideration with this method could be to add a contact form to your site, where customers can request their prescription. Using a contact form, you can gather your customers name, contact information, and request verification of their prescription before creating an order for them.
Let me know what you think about these ideas, and I'll be happy to do some more brainstorming with you! Feel free to reply here with any further thoughts or questions you have.