Yes and No.
It's a common use case for Cloud Pages to build Lead signup pages as you've described. Sending an Email without having a unique ID defined is doable by defining the Email Address as the identifier for that particular subscriber.
However you need to understand that there needs to exist a 1-to-1 link between your Contact and the
AllSubscribers table, meaning if you don't have an ID for your Contact, the Email Address needs be marked as such in order to send Emails.
You'll be utilizing AMPscript InsertData or UpsertData (depending on your exact requirements) to push data from the Cloud Page to your target Data Extension, commonly done via regular HTML POST function.
Generally you would start out by creating a HTML form on your 'Welcome Page' to collect field inputs and posting the form to your 'Thank You' page which holds the short AMPscript snippet to process the user inputted data to the Data Extension.
Example Welcome Page
<form id="myNewForm" method="post" action="yourThankYouPageURL">
First Name <input type="text" name="FirstName" />
Last Name <input type="text" name="LastName"/>
Email <input type="text" name="Email" />
<input type="submit" />
Example Thank You Page
set @FirstName = RequestParameter("FirstName")
set @LastName = RequestParameter("LastName")
set @Email = RequestParameter("Email")
InsertData("Your Data Extension","FirstName",@FirstName,"LastName",@LastName,"Email",@Email)
Thank You ~
This setup requires you to have a Data Extension with these 3 fields.
There is an slightly easier, UI-based way to set it up by using Cloud Pages SmartCapture form that does the heavy lifting for you.
When you finally have some data in your Data Extension, then you can either use Automation Studio (easy), Journey Builder (easy) or Triggered Sends (complex) to send the Email to your fresh lead. Although you will always need to use some field as the
SubscriberKey in order to send an Email.
If your instance comes with Journey Builder, then the easiest and most efficient way would be to utilize it. There is even a specific Entry Source created just for new CloudPage submissions which will work ideally in your use case.