Tira here to help.
Thanks for your question. I would love to dig into some ideas for you and have a few questions that I would like to get your input on so that I can tailor my resources to what you need:
Once I have more information and hear from you, I can help you out with your question.
Thanks for the details, @HCMDigital and thanks @ovicko for offering to help.
Adding a calendar for bookings without using an app would be outside our scope of support. This also sounds like you would need to use a third-party option. I recommend using a search engine like Google and search booking services. I have seen some out there that would provide you an HTML code that you can embed into your website.
Once you have found a code you can use, you can add it into the HTML section of the rich text editor on your product page. Check out our help doc for the steps to do this here: Add HTML content with the rich text editor.
I would recommend moving forward with an app if you can, as this may save you time having to update your HTML each time if you need to in the future, as apps are updated by third-party developers and can be maintained from one section of your store. Here are some suggestions from a similar thread in the Shopify Community: Calendar Functionality.
What do you think? I am also curious to how @ovicko and the Shopify Community can help.