In the example app you provided, focus stays on the email input. You can check it using code below:

driver.get("https://boxing-registration.herokuapp.com/")
email = driver.find_element_by_id("email-input")
email.send_keys("lala")
active_element = driver.execute_script("return document.activeElement")
print(active_element.get_attribute("outerHTML"))
assert active_element == email

You can try to use ActionChains, example:

ActionChains(driver).send_keys_to_element("lala").click(dropdown_list).perform()

I have an HTML <input> field which produces autocomplete suggestions as the user types. I would like to make an automated test where the Selenium driver inputs some keys and then checks the contents of the autocomplete dropdown. The problem is that if I use the Selenium send_keys method, the <input> loses focus and the dropdown disappears. Is there a way to enter keys into an <input> element without losing the focus?

I made a minimalistic example:

from selenium import webdriver

class TestLogin():
    def setup_method(self, method):
        self.driver = webdriver.Chrome()
        self.vars = {}

    def teardown_method(self, method):
        self.driver.quit()

    def test_minimalistic_registration(self):
        self.driver.get("https://boxing-registration.herokuapp.com/")
        self.driver.find_element_by_id("email-input").send_keys("lala")
        print("done")

You can put a breakpoint on the last line and see that the focus of the <input> element is lost.