At this time, HubSpot CRM does not offer embeddable forms for collecting contacts on your site.
HubSpot's Marketing Platform includes forms that can be used to capture contacts. You can try out the feature with a free 30-day trial of the Marketing Platform.
Alternatively, this will require a bit of work for one of your website developers, but you could send contacts from your own forms to the CRM through the Contacts API. You can read through the documentation here.