Hi Simona,
> Web module license is not enough; it will be necessary the "CrossBrowserTesting"
As per https://support.smartbear.com/testcomplete/docs/app-testing/web/cross-platform/index.html#requirements, in TestComplete you will need Web Testing plugin with the Cross-Platform Web Testing sub-component been enabled.
Cross-platform web testing mode in TestComplete uses Selenium to drive tested web browser. So yes, you will need either a license for the cloud service (CrossBrowserTesting, BrowserStack, etc.) or to setup and maintain your local Selenium server (with optional Grid).
P.S. Note, that as cross-platform mode uses Selenium and its locators to locate tested web elements, your script code will require adoption - you will have to remove all native finds provided by TestComplete and replace them with corresponding locators in Selenium style.
To rephrase: old classic TestComplete web tests cannot be executed 'as-is' in the cross-platform mode. Hopefully, this will be improved in future versions of TestComplete.