Yes, WP to Twitter can access any custom field. If the custom field is stored in a custom manner such that you wouldn’t want to use it raw from the database, you can also filter it.
You need to know the meta_key that the custom field is stored under in the database. You’d be able to access it in templates using double square braces:
my_field_key
If you need to customize the content, here’s an example showing how to filter it:
https://github.com/joedolson/plugin-extensions/blob/master/wp-to-twitter/custom-field-content.php
In this case, the field referenced is author_name, but instead of fetching a post meta field named ‘author_name’, the template tag is processed to fetch the first name of the author stored in author meta.