Can I ask customers for extra details, like a company name, when they sign up?

You can create the fields, but the storefront sign-up doesn't save them yet.

To create one, go to Customers → Signup & Login, open the Custom Fields tab and click Add Field:

• Enter a Label and pick a Type — Text, Textarea, Number, Select, Checkbox, Radio or Date. For Select and Radio, add Options (one per line).
• Set the Sort order, and tick Required, Show on signup and Active as needed.

Active fields appear on Add User and on each customer's profile in the admin, and values entered there are saved. Show on signup only shows the fields on page-builder sign-up sections, and what shoppers type there isn't saved yet — for now, add these details from the customer's profile.