Using the app you will see automatically your calendar appointments when participating in a poll or creating a new poll. But you only see your local phone calendar - the calendar(s) you are using with your phone. You don't see the calendars which are only connected to your Doodle account via web.
However, you have to make sure that the local calendars are shown when using the app.
On iOS: You can choose the calendars to show by tapping on your profile picture in the top left corner of the screen. Then under "Settings" go to "Calendars.”
On Android: To choose the calendars to show, tap on the three horizontal lines in the upper left corner. Then go to "Settings". Tap on "Calendars".
Because of technical reasons a direct sync with the calendars connected to your Doodle account via web, when participating or creating a poll is not possible. We are only able to access your local phone calendar(s) but not your connected calendars from your Doodle account although they might show under the "Manage Calendars" section of your settings.
If you want to sync your participation with your calendar, please participate using the desktop version. Or subscribe to the Doodle ICS feed in your calendar. You can find your Doodle ICS feed here (using the desktop version): https://doodle.com/settings/calendars