Hi Kenneth,
Yes, it's possible.
- Add a custom style class (e.g., dropdowSearch--js) to the DropdownSearch widget as highlighted below.

- On Button click, invoke the open method for the dropdownSearch element using JS as shown below.
JS Snippet:
setTimeout(function() {
document.getElementById($parameters.DropdownSearchId).querySelector('.dropdownSearch--js').open();
});
Demo screen: DropdownSearchTask
Note: The DropdownSearch widget in the latest OutSystemsUI references Virtual Select JS library
For more info, refer to the official Virtual Select JS documentation: https://sa-si-dev.github.io/virtual-select/#/methods?id=open
Refer to the attached oml file.
I hope this helps you!
Kind regards,
Benjith Sam