I've modified this topic's title to better reflect your request.
From your description, it sounds like you could still use the Forms API to pass subscriber and customer information into your HubSpot account.
Did you use the v2 forms endpoint or the v3 forms endpoint?
You can use either to pass form information into HubSpot, but only the v3 endpoint supports CORS and AJAX requests.
Alternatively, it is also possible to create and update contact records in your account with the Contacts API or the Ecommerce Bridge APIs. Since neither of these APIs support CORS AJAX requests, however, you would not be able to run them client-side from your website. If you wish to use either of these APIs, you would need to create an intermediary app which could parse the information from your web forms and pass them into HubSpot.
I'll leave the community to respond to your help request, but I also highly recommend reviewing the HubSpot Agency Directory. The partners I linked are all certified with HubSpot and offer integration services.