Is there a way for wholesale customers to sign up for an account and automatically get the wholesale role?

You can, of course, have your wholesale customers sign up via the built-in WooCommerce account registration but as you mentioned it means you have the admin headache of going in and changing the user role on those people』s accounts so they get access.
If you』re looking to streamline this process and make it a bit easier to recruit and manage your wholesalers you might want to take a look at our WooCommerce Wholesale Lead Capture plugin.
It lets you provide your wholesale customers with a special registration form and a special login form and also has user moderation built in so you can approve/deny people as they sign up.
All the emails are also customisable so it gives your incoming wholesale customers a really good experience.
And, it can also be set up to automatically apply the correct wholesale role that you choose.