I have done this before, for a Technology events startup, who needed to limit entries to specific events, and after a certain number of sign ups, inform users that they were being added to a waiting list (or with some events, simply disable further registration).

The solution I chose leveraged gravity forms (a paid plugin) and gravity perks extension that limits entries (a paid extension).

Some more information about this set up is here:

https://gravitywiz.com/documentation/gravity-forms-limit-submissions/

GF does have built-in form submission limiting, and this plugin could be extended exactly how you are looking for (with custom messaging after limits are reached, as well as routing emails to a different list or recipient based on limitation status, or even conditional business rules for submissions based on the limitation status) with some custom PHP and development effort, but I chose the paid plugin because the client was happy to have the issue resolved quickly.

Along these lines, I think just about any form solution could be extended or customized to provide the functionality you’re looking for, and each approach would have advantages/disadvantages relative to your site and business needs, as well as varying levels of time and effort needed.

This is the first time manage an event with a form to sign up. Normally I use “Multi-Step-Forms” to create forms. For me, it’s no option to get a paid Plugin, so I need an option that’s free to use.
Do you know a free Plugin or a workaround for that?

This free plugin includes built-in “entry limitations” functionality that seems to provide the functionality you’re looking for — limiting submissions and displaying a customized notification to users when the form limit has been reached.

Additionally, here’s a step-by-step walkthrough I found for setting this up, originally posted in February of this year (I also checked, the plugin is compatible with the latest version of WordPress with an update just last week):

https://wedevs.com/113912/form-expiration-entry-limitation-wordpress/

When the form is being output, code should query existing registrations to see if the limit has been reached. Depending on the result, the form could be altered to be a waiting list form.

When the form is processed, the registrations should be checked again because it’s possible to submit a form even if it were disabled in the client. Depending on the result, either add a registration or add a wait list item.

If you need specifics on where form data is saved or how to alter form content, I suggest you ask in the form plugin’s dedicated support forum.